De la science à l`application - La Recherche

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Ecole Polytechnique
Département de Mécanique
Laboratoire d’Hydrodynamique de l’Ecole Polytechnique (LadHyX)
Ecoulements diphasiques en microfluidique :
De la science à l’application
Thèse d’Habilitation à diriger des recherches
de l’Université Denis Diderot (Paris VII)
présentée par
Charles N. Baroud
Soutenance le 24 octobre 2008 devant le jury composé de :
Françoise Brochard
Yves Couder
Jacques Magnaudet
Howard Stone
Jean-Louis Viovy
Présidente
Rapporteur
Rapporteur
Rapporteur
Examinateur
Note sur la langue
Ce document est rédigé en anglais car, même si j’aprécie et parle la langue de Hergé
et de Camus, je l’écris mal. J’ai donc décidé d’utiliser un anglais plus neutre et que je
contrôle mieux, pour ne pas offenser la sensibilité des lecteurs francophones qui trouveront
sûrement dans ma prose des erreurs ou des expressions “qui ne se disent pas comme ça”.
Illustrations et films
Une partie importante des images présntées dans ce document, surtout dans les chapitres
2 et 3, sont extraites à partir de séquences vidéo. Les vidéos associées à ces séquences
peuvent être visionnés sur la page web suivante :
http://www.ladhyx.polytechnique.fr/people/baroud/movies.html
Résumé
Ce mémoire présente certains travaux en microfluidique, réalisés depuis 2002 au sein
du LadHyX. Nous commençons, dans le premier chapitre, par décrire le contexte de
la microfluidique de gouttes, aussi bien du point de vue pratique que du point de vue
des questions fondamentales. En effet, si les gouttes microfluidiques représentent une
piste prometteuse pour la miniaturisation des opérations chimiques et bio-chimiques dans
les laboratoires-sur-puce, leur utilisation pose certaines questions auxquelles il faudra
répondre avant que cette utilisation ne puisse devenir généralisée.
Certaines de ces questions concernent l’évolution de la forme et de la vitesse d’une
goutte dans un microcanal, en particulier par rapport au fluide porteur et dans des
géométries typiques de la microfluidique. Quelques résultats sur ces questions sont donnés
dans le chapitre 2, où nous étudions d’abord la stabilité de division d’un doigt de fluide
peu visqueux dans une bifurcation remplie d’un fluide visqueux. Nous démontrons que la
solution symmétrique est toujours instable dans le cas de canaux ouverts à l’atmosphère.
Cependant, l’ajout de membranes élastiques au bout des canaux de bifurcation peut
stabiliser la division symmétrique pour des nombres capillaires assez faibles.
Ensuite nous nous penchons sur le transport de bouchons de liquide poussés dans un
microcanal par une pression d’air. Nous démontrons d’abord une relation nonlinéaire
entre la vitesse d’un pont et l’inverse de sa longueur, pour une pression donnée. Cette
relation peut-être comprise en tenant compte du changement de la forme des interfaces
avant et arrière, dus aux équilibres visco-capillaires au voisinage de ces interfaces. En
particulier, le facteur dominant a pour source l’angle de contact dynamique à l’avant du
bouchon qui introduit une résistance supplémentaire au mouvement. Ce résultat peut
élucider l’augmentation de la résistance au mouvement dans un écoulement diphasique
par rapport aux écoulements mono-phasiques.
Par ailleurs, nous étudions le passage d’un tel bouchon à travers une bifurcation en T
à angles droits, soumis à un forçage à pression constante. Ce passage est caractérisé par
trois évolutions possibles : le blocage du bouchon à la bifurcation, la rupture du bouchon,
ou sa division en deux ponts dans les canaux fils. Alors que la pression de blocage peutêtre comprise à partir de la simple pression de Laplace à l’avant du bouchon, un équilibre
visco-capillaire doit de nouveau être évoqué pour déterminer la transition entre la rupture
et la division.
Enfin, ce chapitre 2 explore aussi la sortie d’un liquide à travers un pore et la relation
entre cette situation et celle de n pores parallèles. Nous mesurons en premier lieu la
forme de l’interface qui tend vers une parabole plutôt qu’un cercle. Nous nous intéressons
ensuite à la sortie de l’interface à travers deux pores parallels; l’évolution de l’interface
dans ce cas montre trois régimes distincts en fonction du temps. Aux temps courts nous
observons une avancée indépendante des deux interfaces, alors qu’aux temps longs nous
3
revenons à la solution parabolique observée pour un pore unique. Cependant, aux temps
intermédiaires, la fusion des deux interfaces qui se rencontrent produit un transitoire
durant lequel le flux augmente, et qui produit une redistribution de l’écoulement vers la
région du centre, entre les deux pores. Cette évolution est ensuite généralisée à n pores où
la redirection du flux vers le centre produit une cascade de fusions en paires de gouttes,
et donc des interactions purement locales.
Le chapitre 3 change de ton et s’attaque aux problèmes technologiques du contrôle
de gouttes microfluidiques par un chauffage local induit par laser. Cette technologie, codéveloppée au LadHyX et brevetée par l’Ecole Polytechnique et le CNRS, peut être utilisée
pour implémenter les opérations de base sur des gouttes microfluidiques, notamment le
contrôle de leur production (fréquence et taille), de leur fusion, division, ou leur tri. Nous
démontrons ainsi comment le chauffage par laser peut-être combiné avec la géométrie du
microcanal pour implémenter toutes ces opérations.
Un modèle hydrodynamique du fonctionnement de cette technique est aussi présenté
afin d’expliquer l’origine de la force agissant sur une goutte. Ce modèle se base sur une
description d’une goutte circulaire entre deux plaques infinies, et résout l’équation de
Stokes tridimensionnelle moyennée dans l’épaisseur. La force prédite par ce modèle montre un scaling qui augmente quand la taille de la goutte diminue, grâce à l’augmentation
de la projection de cette force sur l’axe quand le rayon de courbure local diminue.
Plus loin, des mesures récentes de température, de la dynamique de formation des
écoulements, et du transport du surfactant, sont exposées. Ces mesures expérimentales
commencent à montrer les limites de l’application de la technique en termes de dynamique,
de dépendance sur les fluides et les surfactants utilisés. Enfin, des manipulations avancées
mettant en oeuvre des méthodes optiques holographiques pour produire des spots laser
variables et complexes sont exposées. Ces techniques ouvrent des possibilités nouvelles
pour la manipulation de gouttes à des débits plus élevés et pour implémenter des opérations plus avancées, comme le stockage ou le changement d’ordre des gouttes.
Le quatrième chapitre présente des travaux récents et en cours sur le transport de
sufractant autour de la surface d’une goutte et la production de gouttelettes de taille
inférieure au micron par le phénomène de tip-streaming. Enfin, deux sujets sont rapidement abordés concernant la statistique des écoulements diphasiques dans des réseaux,
ainsi que des applications à la biologie cellulaire du contrôle de gouttes.
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Acknowledgements
This document presents the fruits of almost six years of work which was performed at
LadHyX. In doing this work, I have benefited from the help of many people, both directly
on the scientific aspects, and indirectly on everything else!
On the direct scientific aspects, special thanks should go to Jean-Pierre Delville and
François Gallaire, who both struggled along with me on trying to get to grips with
the laser manipulation problems. More recently, it has been a pleasure to work with
David McGloin and David Burnham on laser heating experiments, Jean-Baptiste Masson,
Antigoni Alexandrou, Pierre-Louis Tharaux, and Paul Abbyad on biological problems.
Benoit Roman and José Bico also frequently contribute to keeping me sain, as well as
Thomas Dubos who teaches me about shear. Patrick Tabeling has also been a source of
inspiration and good ideas.
I have been surrounded by very gifted collaborators in the last years namely Cedric
Ody, Emilie Verneuil, Maria-Luisa Cordero, Yu Song, Michaël Baudoin, who have had
the courage to work in my group. In this, I have also benefited from the help of Paul
Manneville on the recent network problems. Undergraduate students have also been
instrumental in moving things forward. Those include Sedina Tsikata, Samuel Lemaire,
Sabine Mengin, Guillaume Boudarham, Paul Reverdy, Xin Wang, and Pierre-Thomas
Brun. The work that these collegues have produced has been impressive and has taught
me tremendous amounts that I hope I haven’t deformed in this document.
Furthermore, Patrick Huerre and Emmanuel de Langre have played an invaluable
mentoring role, because it’s not enough to have ideas, you have to be able to get them to
work. Thérèse Lescuyer had a huge impact by helping to manage all the material aspects.
Outside LadHyX, I have been enjoying life in common with Antonia, as well as frequent phone calls to Lebanon to my parents and family. I thank them all for the love
they give me.
5
Contents
Curriculum Vitae
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1 Introduction
1.1 Drops in microchannels . . . . . . . . . . . . . . . . . . .
1.1.1 Dispersion . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Mixing . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Manipulation . . . . . . . . . . . . . . . . . . . .
1.2 The questions that drops raise . . . . . . . . . . . . . . .
1.2.1 Motion and shape of clean droplets . . . . . . . .
1.2.2 Presence and transport of surfactants . . . . . . .
1.2.3 Singular events: Formation, merging and splitting
1.3 Manipulating drops . . . . . . . . . . . . . . . . . . . . .
1.3.1 Channel geometry . . . . . . . . . . . . . . . . .
1.3.2 Imposed external flow . . . . . . . . . . . . . . .
1.4 Active integrated manipulation . . . . . . . . . . . . . .
1.4.1 Drops on an open substrate . . . . . . . . . . . .
1.4.2 Drops in a microchannel . . . . . . . . . . . . . .
1.5 Manuscript organization . . . . . . . . . . . . . . . . . .
2 Drops in funny channels
2.1 Flow of a low viscosity finger through a fluid-filled
2.1.1 Linear stability analysis . . . . . . . . . .
2.1.2 Experimental realization . . . . . . . . . .
2.2 Flow of a plug in a straight channel . . . . . . . .
2.2.1 Model . . . . . . . . . . . . . . . . . . . .
2.2.2 Comparison with experiment . . . . . . . .
2.3 Flow of a plug in a bifurcation . . . . . . . . . . .
2.3.1 Blocking . . . . . . . . . . . . . . . . . . .
2.3.2 Rupture . . . . . . . . . . . . . . . . . . .
2.3.3 Splitting into two daughter plugs . . . . .
2.3.4 Modeling the transitions . . . . . . . . . .
2.4 Parallel channels . . . . . . . . . . . . . . . . . .
Two articles . . . . . . . . . . . . . . . . . . . . .
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bifurcation
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3 Droplet manipulation through localized heating
3.1 Manipulating drops with a focused laser . . . . .
3.2 How does it work? . . . . . . . . . . . . . . . . .
3.3 Detailed measurements . . . . . . . . . . . . . . .
3.3.1 Temperature field measurements . . . . . .
3.3.2 Time scale for activation . . . . . . . . . .
3.3.3 Surfactant transport . . . . . . . . . . . .
3.4 What can you do with it? . . . . . . . . . . . . .
3.4.1 Drop fusion . . . . . . . . . . . . . . . . .
3.4.2 Synchronization . . . . . . . . . . . . . . .
3.4.3 Routing . . . . . . . . . . . . . . . . . . .
3.5 More advanced manipulation . . . . . . . . . . . .
Two articles . . . . . . . . . . . . . . . . . . . . .
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4 Recent and future work
4.1 Tip-Streaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Flows in networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Manipulation of single cells in drops . . . . . . . . . . . . . . . . . . . . .
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107
8
Charles N. Baroud
Né le : 23/11/1972
Nationalités : Liban et USA
Email :
[email protected]
Laboratoire d’Hydrodynamique (LadHyX)
Ecole Polytechnique
F-91128 Palaiseau, Cedex
01.69.33.52.61
Formation
– The University of Texas at Austin
Doctor of Philosophy (Doctorat)
12/01
Thèse expérimentale sous la direction de Harry Swinney, intitulée “Transitions from threeto two-dimensional turbulence in a rotating system.”
Master of Science in Mechanical Engineering
01/98
Spécialisation sur les systèmes dynamiques et le design. Etude expérimentale, sous la
direction d’Ilene Busch-Vishniac, intitulée “Induced micro-variations in hydro-dynamic
bearings.”
– Massachusetts Institute of Technology
Bachelor of Science
01/94
Programme combiné en génie mécanique et physique. Mémoire sur la dynamique nonlinéaire et le chaos.
Brevets
• Demande de brevet français No. 0407988 : “Circuit microfluidique à composant
actif” C.N. Baroud, J-P. Delville, R. Wunenburger, et P. Huerre. Déposée le 19
juillet 2004. Étendue à l’international, juillet 2005. Publiée : 01/2006. Dépot
international (Europe, USA, Canada, Japon): 01/2007.
• Demande de brevet français : “Procédé de traitement de gouttes dans un circuit
microfluidique”. C.N. Baroud et J-P. Delville. Déposé en mai 2006.
9
Publications
“Induced micro-variations in hydrodynamic bearings,” C.N. Baroud, I. Busch-Vishniac
and K .Wood, ASME Journal of Tribology, v.122, pp.585-589 (2000).
“Experimental and numerical studies of an eastward jet over topography,” Y. Tian,
E.R. Weeks, K. Ide, J.S. Urbach, C.N. Baroud, M. Ghil, and H.L. Swinney, Journal
of Fluid Mechanics. v.438, pp.129-157 (2001).
“Nonlinear determinism in time series measurements of 2-dimensional turbulence,” M. Ragwitz, C.N. Baroud, B. B. Plapp and H. L. Swinney. Physica D, v.162, pp.244-255 (2002).
“Anomalous self-similarity in rotating turbulence,” C.N. Baroud, B. B. Plapp, Z.-S. She
and H. L. Swinney. Physical Review Letters, v.88, pp.114501 (2002).
“Scaling in three-dimensional and quasi-two-dimensional rotating turbulent flows,” C.N.
Baroud, B. B. Plapp, Z.-S. She and H. L. Swinney. Physics of Fluids, v.15 pp. 2091-2104
(2003).
“Reaction-diffusion dynamics: confrontation between theory and experiment in a microfluidic reactor,” C.N. Baroud, F. Okkels, L. Ménétrier, and P. Tabeling. Physical
Review E, v.67, pp.060104(R) (2003).
“Nonextensivity in turbulence in rotating two-dimensional and three-dimensional flows,”
C.N. Baroud and H. L. Swinney. Physica D, v. 184, pp. 21-28 (2003).
“Multiphase flows in microfluidics – a short review,” C.N. Baroud and H. Willaime.
Comptes rendus - Physique, v. 5, pp. 547-555 (2004).
“The propagation of low-viscosity fingers into fluid-filled, branching networks,” C.N. Baroud,
S. Tsikata et M. Heil. Journal of Fluid Mechanics, v. 546, pp. 285-294 (2006).
“Transport of wetting and partially wetting plugs in rectangular microchannels”, C. Ody,
C.N. Baroud et E. de Langre. J. of Colloid and Interface Science, v. 308, pp. 231-238,
2007.
“Thermocapillary valve for droplet formation and sorting, C.N. Baroud, J-P. Delville, F.
Gallaire et R. Wunenburger.” Physical Review E, v. 75 p. 046302(1-5) (2007).
“Capillary Origami: spontaneous wrapping of a droplet with an elastic sheet”. C. Py,
P. Reverdy, L. Doppler, J. Bico, B. Roman and C.N. Baroud. Physical Review Letters,
v. 98, p. 156103, 2007.
“An optical toolbox for total control of droplet microfluidics”. C.N. Baroud, M. Robert
de SaintVincent, et J.P. Delville. Lab on a Chip, v. 7, pp. 1029-1033, 2007.
“Capillary origami”, C. Py, P. Reverdy, L. Doppler, J. Bico, B. Roman and C.N. Baroud.
Physics of Fluids, v. 19, p. 091104 .
10
“Collective behavior during the invasion of a network of channels by a wetting liquid”,
X.C. Wang, C.N. Baroud. and J.B. Masson. Submitted, 2007.
“Thermocapillary manipulation of droplets using holographic beam shaping: Microfluidic
pin-ball”, M.L. Cordero, D.B. Burnham, C.N. Baroud, and D. McGloin. submitted, 2008.
Expérience professionnelle
Ecole Polytechnique, Palaiseau
2002-présent
• Professeur chargé de cours au département de mécanique depuis 09/2006.
• Maı̂tre de conférences de 2002-2006.
Chargé de démarrer l’activité en microfluidique au LadHyX.
Ecole Nationale Supérieure des Techniques Avancées (ENSTA)
Enseignant vacataire, cours d’ouverture en Microfluidique.
2005-2007
Ecole Normale Supérieure, Paris
2001-2002
Séjour post-doctoral en microfluidique avec Patrick Tabeling. Méthode d’analyse des réactions chimiques en microfluidique, et mélangeur chaotique. Encadrement de stagiaires.
Bourse du ministère de la Recherche.
The University of Texas at Austin
Center for Nonlinear Dynamics
1996-2001
Thèse : mise au point et exploitation d’une plaque tournante afin d’étudier les écoulements turbulents en rotation. Conception et mise au point d’un système de visualisation
numérique (PIV). La transition entre un écoulement tri-dimensionnel et bi-dimensionnel
est observée par des mesures statistiques (séries temporelles) et dynamiques (champs de
vitesse) en fonction de la vitesse de rotation.
UT-MEMS group
1996
Mesures de l’effet de changements micrométriques de la surface d’un roulement hydrodynamique sur les forces exercées.
Sun Microsystems
1994-1995
Responsable du support technique francophone pour l’Amérique du Nord, pour tous les
produits hardware et software de Sun Microsystems.
Massachusetts Institute of Technology
1990-1994
Plusieurs projets de recherche dans les départements de physique et de génie mécanique;
travail expérimental sur le projet LIGO; numérique sur les systèmes Hamiltoniens.
11
Etudiants au LadHyX
• Cédric Ody : Thèse (2004-2007) co-encadré avec Emmanuel de Langre.
• Maria-Luisa Cordero : Etudiante en thèse (octobre 2006- ).
• Emilie Verneuil : Post-doctorante (septembre 2006- août 2007)
• Michaël Baudoin : Post-doctorant (2008)
• Yu Song : Etudiante en thèse (octobre 2007- ) co-encadrée avec Paul Manneville.
• Environ 10 stagiaires de l’Ecole Polytechnique, de l’ENS, de master, ou d’universités
étrangères (MIT, U.C. Berkeley, U. of Saint Andrews).
Activités académiques
• Membre du jury de thèse de Marie-Noelle Fiamma, UPMC, novembre 2006.
• Organisateur du mini-symposium microfluidique au European Conference on Fluid
Mechanics, juin 2006.
• Membre du comité du département de mécanique, 2004-2007.
• Reviewer pour les journaux suivants: Physical Review Letters, Journal of Fluid Mechanics, Physics of Fluids, Lab on a Chip, Microfluidics and Nanofluidics, Journal
of Applied Physiology, Journal of Micromechanics and Microengineering, Comptes
Rendus de l’Académie des Sciences.
Séminaires et conférences invités
1st European Conference on Microfluidics, Bologna, Italie
Conférence invitée.
12/08
Microdroplets in action symposium, University of Cambridge
05/08
Laboratoire MMN, ESPCI, Paris
Séminaire du laboratoire.
06/07
Journée Microfluidique, Paris
05/07
University of Saint-Andrews, Ecosse.
Colloquium du département de physique
03/07
Centro de Investigación en energia, UNAM, Mexico
Seminaire du “departamento de termociencias”
02/07
University of Cambridge, Department of Chemistry, Angleterre
Séminaire du “microdroplet group”
02/07
12
Workshop ATOM3D, Thales Research & Technology, Palaiseau, France
Présentation invitée
07/06
Institut Català de Nanotecnologia, Barcelone
Séminaire de l’institut.
06/06
Université Joseph-Fourrier, Grenoble
Séminaire du laboratoire de Spectroscopie
05/06
Laboratoire d’Optique de Biosciences, Ecole Polytechnique
05/06
Ecole Normale Supérieure de Cachan, Antenne de Bretagne
Séminaire du groupe de mathématiques.
04/06
University of Cambridge, Angleterre
02/06
• Séminaire du BP Institute for Multiphase Flow
• Séminaire du Department of Applied Maths and Theoretical Physics (DAMTP)
University of Bristol, Angleterre
Séminaire du groupe de mécanique des fluides, département de maths
02/06
University of Texas at Austin
Séminaire du Center for Nonlinear Dynamics
11/05
Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), Marseille
Séminaire du laboratoire
12/04
Université de Lille I, Lille
11/04
Séminaire inter-laboratoires physique (CERLA) et chimie (Chimie Organique et Macromoléculaire)
University of Manchester, Angleterre
Séminaire du Manchester Centre for Nonlinear Dynamics
02/03
Observatoire de Paris, Meudon
Séminaire de l’observatoire
05/02
Université Paris Sud, Orsay
Séminaire du laboratoire FAST.
03/02
Ecole Normale Supérieure, Paris
Séminaire du Laboratoire de Physique Statistique.
01/02
American University of Beirut
Séminaire du departement Mechanical Engineering.
12/01
Laboratoire Coriolis, Grenoble
01/01
13
University of California, Berkeley
Séminaire geophysical fluid dynamics.
10/00
University of Texas at Austin
Séminaire du groupe Thermo/Fluids.
10/99
American University of Beirut
Séminaire du Center for Advanced Mathematical Studies.
06/99
Interventions orales et posters, après 2000
Zing conference, Cancun, Mexique
Laser control of microfluidic drops.
02/08
MicroTAS 2007, Paris
10/07
Flow and temperature field measurements in a flow focusing geometry subject to optical
forcing
M.L. Cordero, E. Verneuil, and C.N. Baroud
MicroTAS 2007, Paris
Giant deformations and tip-streaming from sheared drops
S. Molesin and C.N. Baroud
10/07
Microfluidique 2006, Toulouse
Thermocapillary manipulation of microfluidic droplets: Theory and applications
C.N. Baroud, J-P Delville et François Gallaire
12/06
MicroTAS 2006, Tokyo
Control of droplets in microchannels through laser-induced thermocapillarity
C.N. Baroud, J-P Delville
11/06
World Congress of Biomechanics, Munich
Modeling pulmonary transport of liquid plugs in microfluidic channels
C. Ody, E. de Langre, et Charles N. Baroud
07/06
APS, Division of Fluid Mechanics meeting, Chicago
11/05
Localized laser forcing for breaking, sorting, or blocking droplets in a microchannel
C.N. Baroud et J-P. Delville.
Euromech 472, microfluidique. Grenoble
Communication invitée
09/05
Gordon conference: Physics and Chemistry of Microfluidics, Oxford
Laser actuated building blocks for droplet microfluidics
C.N. Baroud et J-P. Delville.
09/05
14
Congrès français de mécanique 2005, Troyes
09/05
Modélisation en microfluidique du transport de ponts liquides dans l’arbre pulmonaire
C. Ody et C.N. Baroud.
SPIE (International Society of Optical Engineering) annual meeting, San Diego
Laser-actuated microfluidic building blocks
C.N. Baroud, J.P. Delville, R. Wunenburger.
08/05
2ème Congrès SHF “Microfluidique”, Toulouse
Pompe et vanne microfluidiques actionnées par laser
C.N. Baroud, J.P. Delville, R. Wunenburger et P. Huerre.
12/04
International Congress on Theoretical and Applied Mechanics (ICTAM)
How to breathe in a liquid-filled lung: symmetry of airway reopening.
C.N. Baroud and M. Heil.
08/04
American Physical Society, Division of Fluid Dynamics, USA
A microfluidic model for lung airway reopening.
C.N Baroud and M. Heil
11/03
MicroTAS 2003, Lake Tahoe, USA
Measurement of fast kinetics in a microfluidic system
C.N Baroud, L. Ménétrier, F. Okkels and P. Tabeling
10/03
Congrès Francais de Mécanique, Nice
La microfluidique appliquée aux systèmes de réaction-diffusion
C.N. Baroud, L. Ménétrier, F. Okkels and P. Tabeling
9/03
Congrès SHF “Microfluidique”, Toulouse
Analyser une reaction chimique dans un micro-Te
C.N. Baroud, L. Menetrier, T. Colin, F. Okkels et P. Tabeling
12/02
Premier colloque Dynamique et Applications, Beyrouth
Micron-scale fluid dynamics applied to fast chemical kinetics.
10/02
American Physical Society, Division of Fluid Dynamics
Measurements of 2D turbulent spectra in experiments on a Beta-plane.
C.N. Baroud, B. B. Plapp and H. L. Swinney.
11/00
Nonlinear-Dynamics and Pattern Formation conference
Jupiter revisited: New images of rapidly rotating turbulence.
C.N. Baroud, B. B. Plapp and H. L. Swinney.
06/00
15
Chapter 1
Introduction
17
Microfluidics is not a science; it is a tool that allows, through the manipulation of
small quantities of fluids, the invention of new and exciting ways to approach problems in
biology, chemistry, nanotechnologies, and many other fields. For this reason, the most important publications and conferences on microfluidics bring together scientists from many
of the fields cited above, in addition to the fluids engineers and physicists. This makes the
field extremely stimulating and encourages people to collaborate together because, unlike
more traditional areas, no one can claim to master all of the required expertise to truly
produce innovative work in microfluidics. A principal role of fluid mechanics research is
therefore to develop further the tools that allow the micromanipulation of fluids and to
understand the limitations and opportunities that emerge.
However, microfluidics is also a way to access fluid dynamical phenomena that are
difficult or impossible to observe by other techniques. The small scales involved imply that
the Reynolds number is usually small, even for moderate fluid velocities (see Ref. [108] for
an excellent review). In this way, one can use microfluidics to access situations in which
viscous or capillary effects are naturally dominant compared to inertial effects. This idea
is not fundamentally new, it follows the same approach as working in the Hele-Shaw
geometry or in capillary tubes. The novelty resides in the ability to make the fluidics
with complex and well controlled geometries. In this case, microfabrication is the tool
that allows new fluid dynamics to be explored.
In both of the cases above, the new techniques also raise significant questions that
have constituted the main body of research in microfluidics. Indeed, since the early
work on the miniaturization of chemical analysis, it has become clear that a broad range
of issues had to be addressed before the devices that were initially promised become
available. This is true for example for biochemical sensing techniques when the number
of molecules becomes small [71], for material compatibility issues when new materials are
used for micro-fabrication [78], as well as for issues in the physical and flow aspects such
as mixing without turbulence [92]. The initial microfluidic devices, which were imagined
by electronicians and chemists, rapidly ran into the limitations of using simple fluidic
resistor models to describe the flow in the channels. One had to realize, for example, that
the parabolic profile of a pressure driven flow had important consequences on the device
functionality [73, 67].
Moreover, multiple attempts to produce new devices by simply reducing the scale
of macroscopic objects have shown that this is often a poor strategy that produces unsatisfactory results. One can see, for example, the decades spent on miniaturizing positive
displacement pumps through complex multi-stage lithography techniques and noticing
that none of these pumps are commercially available [76]. This is because the force
balance is often changed due to miniaturization even though the continuum description
remains valid to remarkably small scales. For instance, a valve that is engineered for the
centimeter scale relies on the implicit assumption that adhesion forces are negligible compared to the inertia of the valve or fluid. However, such a valve may become unusable, as
the scale is reduced, if adhesion is not accounted for. The often quoted adage that surface
effects begin to dominate over volumetric effects is true, even though the implications of
this domination are not always clear a priori.
Finally, one cannot forget the limitations imposed by microfabrication techniques.
The vast majority of microfluidics research is done in channels that use some form of
photo-lithography and surface etching. Although these techniques allow for the fab18
rication of almost any planar geometry, they rapidly run into difficulties when threedimensional structures are needed [85]. This implies that making a three-dimensional
serpentine channel for chaotic mixing [83] is too difficult to be widely used. More importantly, it implies that most microchannels have a quasi-rectangular cross-section. So
forget about all of the axisymmetric analytical solutions to the Stokes flow equations.
All this motivates the fluids physicist or engineer to explore the fluid dynamical
aspects of microfluidics. The benefit from this is two-fold:
A – Important advances can be made in understanding the behavior of fluids and beautiful phenomena, which may only be observable on small scales, can be discovered [108]. This is true of many Stokes flow problems that are easier accessed on
small scales [67, 13, 10], as well as issues related to flows of rarefied gases when the
mean free path becomes comparable to channel size [9, 87], or interactions of viscous
flows with deformable membranes such as biological cells [2]. However, nowhere are
these new phenomena more present than in multiphase flows, where the interplay
between surface tension, viscosity, and geometry produces a plethora of new and
surprising situations.
B – An understanding of fluid mechanical concepts opens the door to the invention of
tools that are more creative and better adapted to the tasks at hand. In particular,
knowledge of certain analytical solutions to fluid flows and can lead to elegant design
solutions. A few striking examples include a passive chaotic mixer [111], a passive
particle separator [65], a temperature cycling device based on thermal diffusion [22],
or a pump that uses nothing but the Laplace pressure to move fluids in a controlled
manner [17]. In all of these devices, complex operations are performed by intelligent
system design rather than by software or complicated actions.
1.1
Drops in microchannels
One such bright idea, consisted of using droplets as mobile reactors inside microchannels,
as shown in Fig. 1.1 [107, 106]. In this figure, three aqueous solutions are introduced
through the top three branches of the channel, while an oil solution is introduced from
the left side. The water at the junction breaks up into separate drops due to the shear
induced by the oil flow. The oil remains a continuous phase due in part to the geometry
of the channel and in large part to the relative wetting properties of the water, oil, and
channel walls. Furthermore, if the three aqueous inlets contain reacting species, these
will react as they come into contact inside each of the drops. In this way, a large number
of microreactors are produced in series and following one drop corresponds to observing
the reaction’s evolution in time.
This idea was not developed in the microfluidics community, it had already been
explored in emulsions in order to “compartmentalize” reactions inside many small parallel
volumes [115]. For Tawfik and Griffiths [115], each drop in an emulsion represents an
artificial cell which can be characterized by a genotype and a phenotype. In other words,
each drop contains a strand of DNA which is allowed to produce the associated proteins
during the course of an experiment. If the original DNA population is produced with
some controlled variations, each drop will be slightly different from its sisters and will
19
Angewandte
Chemie
Communications
Aqueous droplets of 250 pL formed in a microfluidic channel in a
continuous flow of a water-immiscible fluid act as microreactors that
mix the reagents rapidly and transport them with no dispersion. These
droplets may also be used to control chemical reaction networks on
millisecond time scale. For more information see the following
publication by R. F. Ismagilov et al.
Figure 1.1: Drops used as continuously flowing micro-reactors, carried by an inert oil
flow, from [107].
Angew. Chem. Int. Ed. 2003, 42, 767
! 2003 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1433-7851/03/4207-0767 $ 20.00+.50/0
767
contain a different population of proteins which can later be measured. In this way, the
simple passage from a continuous solution to a large number of independent droplets
multiplies the total number of experimental realizations, as long as the droplets are well
isolated from each other by the continuous phase.
This approach is only useful if two requirements are met: The first is that the volume
of each drop is adapted to the experiment that one wishes to perform and the second is
that the drops can be manipulated (formed, tested, sorted, or stored) individually. It is
therefore a lucky strike that drops measuring a few tens to a few hundreds of microns are
well adapted to a large range of experiments and that they can be manipulated rapidly
and robustly in microfluidic systems!
By the same token, the use of drops addresses some of the most fundamental problems in microfluidics: dispersion, mixing, and the manipulation of small volumes, as
detailed below.
1.1.1
Dispersion
The most obvious problem that is circumvented by drops is the dispersion of species
through Brownian diffusion and Taylor-Ariss dispersion, both of which act to reduce the
concentration of the flowing species. In the case of droplet microfluidics, a molecule (or
cell, or particle) contained inside a drop will remain localized inside the drop, as long as
the fluids are properly chosen such that the species is not soluble in the continuous phase.
In this way, the initial concentrations inside each drop can be maintained throughout the
length of the experiment.
This general statement is only true if care is taken in the choice of fluids because
20
even very weak miscibility can induce large effects due to the very small volumes at play.
For instance, the water contained in the drop can dissolve in the oil phase, due to weak
miscibility, if the oil is pure and if the drops are small enough. This effect has been used in
order to vary the concentration of a solute in a drop by actively changing its volume [68].
Also, intermediate species during a chemical reaction can be miscible in the oil even if
the starting and ending species are not. This may cause problems in the case of complex
reactions which rely on a large number of intermediate reaction products. Finally, some
species can show micellar solubilization, meaning that the molecules will get trapped in
the surfactant micelles even if they are not soluble in the oil itself [121].
However, many systems have already been shown to produce satisfactory results
and a few fluid combinations have emerged as standard and robust. While water is the
droplet liquid of choice for biological applications, fluorinated oils have been shown to
have many attractive features that make them a good carrier fluid for biological samples.
They have been successfully used in both molecular and cellular systems [125, 32, 25]. In
contrast, silicone oils have also been used to study nucleation kinetics of salts in water
drops [77]. Finally, alcanes are often used as cheap and easy substitutes for many fluid
mechanical studies [46, 7, 82].
1.1.2
Mixing
Mixing in the absence of turbulence has challenged scientists since the early days of
microfluidics. Early on, flows exhibiting Lagrangian chaos were considered for mixing
and both passive and active mechanisms were developed for single phase flows, in order
to produce the required stretching and folding of material lines [83, 111, 92]. Immiscible
interfaces can readily be used to induce flows that display Lagrangian chaos, for example
by flowing drops through the zig-zag pattern shown in Fig. 1.1. Here, the presence of
the immiscible interface modifies the flow lines, creating recirculation rolls in each fluid
which are aligned with the instantaneous direction of the drop. They therefore change as
the drop direction is changed in the zig-zag, thus providing a robust way of separating
initially close particles [107, 102, 103].
Note however that most of the studies on mixing have been done for ideally shaped
drops or bubbles, having several planes of symmetry. However, drops and bubbles can
have many different shapes and it has been shown that small changes in the drop shape
can strongly modify the mixing efficiency. Shape anomalies can enhance the mixing even
in a straight channel [107] but they can also create dead zones inside the drops in which
the fluids do not mix [102].
A related effect involves residence time of a fluid particle in the channel. Indeed, the
recirculation rolls due to the interface force the liquid from the channel center to the wall
regions and vice versa, as shown in Fig. 1.2. Therefore all fluid particles will go through
periods in the slow flow regions followed by periods in the fast regions, thus cancelling
the variations due to the parabolic flow profile [61]. This “mixing of velocities” is of major
importance in the synthesis of materials, for which a good control of the reaction time is
crucial in the outcome of the reaction [72].
21
and nanoparticles have been sh
along gas–liquid–solid contact
film and menisci locations can
deposition patterns during the
nanoparticles in segmented g
electron micrograph obtaine
(Fig. 10e) after a particle sy
microchannel bottom wall and
ferentially deposit in the film r
gas–liquid–solid contact lines p
menisci regions are, however, alw
particles deposited in microcha
much smaller.
The mechanisms through whi
are interconnected influence the
and hence the residence time (R
flow. Trachsel
Figure 1.2: Recirculation rolls measured by micro-PIV between two air bubbles,
as dis- et al. measured
through a rectangular cross sec
cussed by Gunther
Jensen patterns,
[61].
Fig. 9and
Streamline
velocity vector field and pressure dislow capillary number flows (Ca
tribution for segmented flow at low Ca and Re numbers. (a) Streamline
116
for particle synthesis.50 RTD
pattern proposed by Taylor
for small relative slip velocities (U 2
Ud)Ud 2 1. (b) 2D computation of streamwise pressure jump at the
reported in capillaries with cir
1.1.3 Manipulation
bubble tip for developed segmented gas–liquid flow through a square
hydraulic diameter but for at
channel.123 (c) Laplace pressure jump across the interface, DpL, during
higher
capillary
numbers.131–133
Finally, a moredroplet
subtleformation
advantage
of
using
droplets
is
in
reducing
the
gap
between
the
for a water and an oil stream joining at a T-junction
dispersion analysis that is bas
79
macro-scale lab postulated
machinesbyon
the one
hand,
as syringes
Garstecki
et al.:
the such
discontinuous
water and
phasetubes
enters that can be hanTheir analysis is based on
dled by the user,
the channel
microchannels
the other
The issue
of connecting
intoand
the main
(I), blocks on
the main
channelhand.
(II), a droplet
is
Thulasidas et al.133 for bubbly
formed,volume
elongatesis
downstream
(III), and
the inlet
(IV).
microchannels whose
smaller than
0.1separates
µL to afrom
syringe
and
tube whose volume
laries. Dispersion results obta
Micrographs
wereimportant
obtained in aissues
33 mm in
deep
andability
100 mm to
widecontrol
main the
is larger than 200
µL raises
the
flow in the
25
display a pronounced tailing eff
channel and a 50 mm wide side channel for 8 6 10 , Ca , 8 6
microchannels. In23particular, very minor changes in the tube cross-section
leadtotothe more Gaussia
in can
contrast
10 . (d) First mPIV measurement of 2D velocity field inside a liquid
major effects onslug
themoving
flow through
in the achannel
so
that
changes
in
flowrate
must
be
made
istics of low on
Ca microfluidic m
rectangular microchannel by Günther et al.
relatively long time
scales.
(Ca = 0.001, Re # 1, Lslug # 2 dh); Ud was subtracted in order to show
partly be attributed to the re
the recirculation
motion insidebe
theused
slug.50,127
considering
the high velocities, f
Using microdrops
can sometimes
to circumvent such difficulties
by keeping
the external forcing constant but using on-chip manipulation of drops. This would in effect
1496of| the
Lab manipulation
Chip, 2006, 6, 1487–1503
This journal is ! The Ro
move the problem
away from the large machines to more precisely
tuned micromanipulation tools [33, 5, 4, 96, 12, 11]. The theme of droplet manipulation
has been one of the major axes of research in our lab, and will be discussed in detail in
Chapter 3.
1.2
The questions that drops raise
As we have seen above, the introduction of drops in microchannels provides an elegant
solution to some of the main problems encountered in continuous flow setups. However,
this comes at the price of raising a new set of fluid dynamical problems that are due to
the deformable interface, the need to take into account surface tension and its variations,
in addition to singular events that appear during the merging or splitting of drops.
One approach to dealing with these complications consists of using the tools at
our disposal to minimize their practical impact. This is typically the approach taken
for chemical applications where the drop is simply a vehicle for transporting reagents.
A complementary approach is to tackle the questions in order to elucidate their mechanisms, possibly with the aim of taking advantage of the new understanding for future
applications. This is the fluid physicist’s niche, one in which innovation can take place.
22
Interface
1.2.1
Motion and shape of clean droplets
Verneuil, Cordero, Gallaire, Baroud
Fundamentally, the moving interface introduces new boundary conditions on the fluid
flow equations. These conditions add a kinematic and dynamic condition which continually impose a movement of this interface and therefore a new geometry. This coupling
between the geometry and the flow constitutes a significant difficulty in looking for analytical solutions to flows of bubbles and drops. It has motivated a large body of research
focused on the limit of spherical or nearly spherical drops, starting with the work of G.I.
Taylor in 1932, where the underlying symmetries can be used to obtain the flow profile
analytically [116, 124]. When the shape departs significantly from sphericity, the recourse
is generally experimental, even for G.I. Taylor who moved on to experiments in 1934 in
order to study
large deformations
drops [117].
The same
coupling
between the flow vos idees s
Quelques
elements
(je vaisofyhisreflechir
mieux,
c’est
promis)(toutes
and the geometry has also hindered advances in numerical simulations of drops since they
Applying
a force
on drop through
effects:
[1] evolution at
generally require
the re-generation
of adaptiveMarangoni
meshes which follow
the drop
each time step
and thus
become computationally
Applying
a local
heating
with a laser expensive.
[2]
Here, we will
significantly
Our previous
study
[3] limit our focus to a subset of the general case. We will
concern ourselves with drops (or bubbles) flowing inside confined geometries, and particFocus
of the present
study:The
measure
nN
forces
applied
on a drop in microfl
ularly rectangular
microchannels.
drops weof
will
consider
here
have a characteristic
length larger and
than the
width
height of theat
channel
and the
are formed
transported
timescales
thus
theandfrequency
which
effectand
can
be usedin for applica
the
microsystem.
The
formation
can
be
made
at
a
T-shaped
junction
by
flowing
into
in favor of a solutal Marangoni origin of the rolls; test thewater
relevancy
of the
one side and oil into another, as shown in Fig. 1.3. This image shows the typical shape
between
thewerolls
and thein,force.
of drops that
are interested
flowing from left to right in a typical microchannel.
November, 16 2007
1
Introdution
Qwater
(2)
Q oil
l
Qoil(1)
500 µm
Figure 1.3: Water drops are formed in a Hexadecane phase at the first T junction, while a
second oil inlet is used to control the total flowrate and spacing between successive drops.
Test sectio
Figure
1: setup
We will also place ourselves in the limit where viscous
effects
are small compared
to surface tension effects. This balance can be quantified through the capillary number
Ca = µU/γ, where µ is the outer fluid viscosity, U its characteristic velocity, and γ the
interfacial tension. In the limit of small viscous effects, i.e. Ca 1, capillarity implies
that the interface shape is a spherical cap.
The first question, whose answer is surprisingly difficult, is to know the relative
speed of the drop with respect to the outer fluid. The schematic view of Fig. 1.4, for
the case of a channel with square cross-section, begins to show the difficulty of knowing
the relative velocities. This is first due to the presence of corners that the drop cannot
Microchannels
- The fabrication of the microchannels is performed using the
fill properly, and through which the outer fluid can flow (called “gutters” in Ref. [56]).
raphy
technique
and velocities
for a full
description
of thefluid,
process
reader can re
Depending
on the relative
of the
drop and the carrier
the flowthe
in these
2
Material and methods
negative imprint (a mold) of the channel is microfabricated in a negative p
23
crochem) by conventional photolithography.
A negative replica of this maste
polydimethylsiloxane (PDMS), a heat curable silicone elastomer (Sylgard 1
65 C for 4 h (the thickness of the elastomer block is of the order of 5 mm
PDMS replica is peeled off and holes are drilled at the channel entrances and
corners can be either forward or backward, in the reference frame of the drop. The
effect of this switching of the flow direction strongly affects the recirculation patterns in
AR266-FL38-11
ARI
23 November 2005
17:2
the inner
and
outer fluids,
thus also changing the mixing and transport characteristics.
Indeed, continuity of velocities on the interface implies that the recirculation rolls inside
the drop must be different if the flow in the corners is faster or slower than the drop.
Annu. Rev. Fluid Mech. 2006.38:277-307. Downloaded from arjournals.annualreviews.org
by Ecole Polytechnique - Palaiseau Cedex on 07/16/07. For personal use only.
Figure 6
Pressure-driven motion of
bubbles in square
channels. (a) The shape at
small C showing the
entrained films: In Wong
et al. (1995b), half of the
bubble is shown. (b) An
illustration for the
intermediate-C-model of
Ratulowski & Chang
(1989). (c) Numerical
(squares) and asymptotic
(solid lines) results for the
pressure drop across the
bubble tip for two
different aspect ratios (α,
not to be confused with
the quantity in Figure 3
and Table 2). From Hazel
& Heil (2002). Reprinted
with permission from
Cambridge University
Press.
Figure 1.4: Shape of a flowing drop in a square channel, from Ref. [6].
Furthermore, the interface shape can vary as a function of forcing pressure, as shown
in part b of Fig. 1.4. Even though this interface retains its spherical shape at the end
caps, the radius of curvature will be different at the front and rear, giving the drop the
“bullet” shape, also seen in Fig. 1.5. This asymmetry implies that the Laplace pressures
at the front and rear of the drop are not balanced, which introduces a supplementary
driving term on the drop in addition to the external forcing.
For sufficiently long drops, the magnitude of the curvature change at the front
interface is determined by Bretherton’s law [24], adapted to the rectangular geometry.
This law predicts the thickness of the deposited film, based on a capillary-viscous balance
at the transition
region between the flat interface near the wall Ajaev
and Homsy
290
· the curved cap at the
front of the drop. In the case of an semi-infinite bubble in a circular tube, Bretherton’s
law predicts that the thickness of the film (e), normalized by the tube radius (R), should
scale as e/R ∼ Ca2/3 .
As for the rear interface, a modified version of Tanner’s law [114] can be used to
predict the change in curvature, based on a visco-capillary balance in the corner flow
between the drop and the solid wall. The standard result is that the advancing contact
angle (θa ) will increase with increasing capillary number as θa ∼ Ca1/3 . The net effect
of the change in the curvature of the two interfaces is to yield a nonlinear relationship
24
oil
(a)
(magnification and numerical aperture). The optical system
expansion
10 mm
focuses the particles that are
within the depth
of focus of the
imaging system, whereas the remaining particles are unfocused
waste
and contribute to the background noise level. Thus, it is important to characterize exactly how thick the depth of correlation
pressure
is. First, we measured
it experimentally, as described in Meintap
22
hart et al. (see Appendix). According to our results, it can be
concluded that the depth of correlation of the micro-PIV system is 9 "m. This is in very good agreement with the theoreti(b)
(c)
Velocity and flow rates
Figure 6 shows the computed v
is fully developed in a time of the
ble 2 shows the comparison betw
velocity and flow rate. The compu
#
Z "Z
I L
ðI # CÞU & ndS
Qc ¼
L 0
S
Figure 1. Schematic of the device. Three devices were studied with cross-sectional dimensions of the main channel (width × depth) being
105 × 74, 200 × 95 and 300 × 95 µm. The water and oil inlets had the same dimensions as the main channel. The pressure taps had widths
of 200 µm in all three devices. The section labeled ‘expansion’ was made by taping Scotch tape onto the SU8-on-silicon mold. Hence it has
greater depth than the rest of the device. The three photographs are (a) a water drop being formed at the T-junction, (b) a single file of drops
in the pressure-measurement section and (c) drops retracting into spheres in the expansion section.
Qd ¼
Table 2. Comparison of the
procedures [15]. The PDMS replicas were exposed to plasma the receding front of the drops suggests that the drops do not
Macrosco
treatment using a Harrick scientific plasma cleaner (PDC- make direct contact with the walls, but instead, a thin wetting
as
32G) at a maximum power setting of 100 W for 30 s. layer of oil remains in contact with the walls at all times,
Parameter
They were then immediately sealed against glass slides. also reported by others [16–18]. This is also consistent with
Silicone
velocity, Uc (m/s)
PDMSoilsheets
External connections to gas-tight syringes were made using our own measurements of the wettability of flat
Silicone oil flow rate, Qc (m3/s)
Figure
5. Shapeouter
of droplets.
medical grade polyethylene
tubing (Intramedic,
diameter coated with OTS using a similar procedure as above (5 min
Droplet velocity, U (m/s)
−1
Figure
1.5:Syringes
Advancing
drops
take
a bullet
shape.
Top
three
show
solution
of OTS).experimental
The coated sheets d
dipping
into
a 0.05
mol limages
0.965
mm).
were driven
by
programmable
(a) two
Experimental
(yz) view
imaged
with
a high-speed
camera
Droplet flow rate, Qd (m3/s)
monitored
on a Velankar
microscope.
(b)
andBottom
(c)to3Dbecomputation.
fully
hydrophobic
(the
oil/water
contact
syringe
pumps. The
wettability
the
microchannel
walls were
photographs
taken
fromofAdzima
and
[3].found
three
images:
Part
(a) is
an
was found to affect the range of flow rates over which drops angle was 180◦ ), consistent with the presence of a wetting oil
experimental image,ofwhile
parts
(b) and (c) represent
numerical
solutions of the drop
on the walls
all times.
could be generated, the size
and10.1002/aic
their behavior layer
4066 the drops,
DOI
Published
on at
behalf
of the AIChE
December 2006 Vol. 52
shape,
from Sarrazin
et al.
[102].
The pressure drop was measured using differential
in
the microchannels
(elaborated
further
at the end of this
paper). Therefore, the following procedure was used to ensure pressure sensors (Honeywell 26PC series) connected across
consistent wetting properties: the microfluidic devices, after pressure taps. The output of the pressure sensor was recorded
Hz on aMore
computer
using a Labview
data acquisition
connecting
but before
passingand
any the
fluid velocity
through atof1000
betweenthe
thetubing,
forcing
pressure
a drop.
practically,
in means
that
the device, were treated again in the plasma cleaner for system. This measured pressure drop corresponds to several
the presence of a drop increases the resistance to flow in a given channel, compared to a
30 s. Immediately after this treatment, a ∼0.05 mol l−1 drops (7–30, see below) that are present in the 10 mm test
single ofphase
flow [55, 104].
solution
octadecyltrichlorosilane
(OTS) in hexadecane was section between the pressure taps. One may expect that as
each drop of
passes
tap, the
may fluctuate
pumped In
intoour
the channels,
allowed
sit for 5treated
min, and then
work [91],
weto have
a reduction
thisa pressure
problem,
by pressure
considering
the
flushed out with pure hexadecane. Extensive past literature [19]. However, in our experiments, this fluctuation was not
motion of plugs of liquid surrounded by air in a microchannel. These plugs can be taken
indicates that the OTS can silanize the plasma-treated PDMS evident, and only the average pressure drop for the two-phase
to represent
space
the drops,
instance
the
white parts separating the
droplet
flow could
be measured.
surface,
resulting inthe
grafting
of thebetween
octadecyl chains
onto the for
Just
before
exiting
the drops
were allowed
surface.
The
resulting
coating
of
OTS
ensured
consistent
gray bubbles in Fig. 1.5(b); this analysis allowed us to derive the
an device,
analytical
formula
for
hydrophobicity of the walls of the device and reproducible to expand into a larger ‘cavity’ that was both deeper as well
the nonlinear relation between pressure andasflowrate
and will be treated in detail in
wider than the microchannels (figure 1(c)). This allowed
behavior of drops in the channels.
Section
2.2.
Experiments were conducted using hexadecane with the drops to relax into spherical shapes, thus allowing their
1 wt% of surfactant Span 80 as the continuous phase and diameter (designated 2Rd) to be measured accurately. The
water as the drop phase. This hexadecane/surfactant mixture single-file flow of drops in the pressure-measurement section
also imaged directly (figure 1(b)), and the spacing between
is1.2.2
henceforth called
‘oil’ in this paper.
was foundofwas
Presence
andHexadecane
transport
surfactants
to swell the PDMS, with an equilibrium swelling estimated the drops was measured from such images. The number of
the test
section was
calculated
from the inter-drop
atTo
about
wt%. However,
this swelling
relatively
rapid
this14already
complex
pictureis one
must
adddrops
the invery
complex
effect
of surfactants
[81].
and was expected to equilibrate before experiments were spacing and was found to range from 7 to 30, depending on
The
vast
majority
of
droplet
microfluidics
systems
use
some
surfactants
in
order
to
stabiinitiated. The swelling also introduces an approximately flow rates. All imaging was performed using an inverted
lizeerror
the inwetting
conditions,
soften
theonly
interfaces,
and to(Olympus
preventCKX41)
unwanted
mergings
the
and a digital
cameraof(Basler
5%
the channel
dimensions;
however,
the microscope
A302fs).
unswollen
used in all calculations.
None
drops ordimensions
bubbles.areGenerally,
surfactant
is ofadded
in large quantities, orders of magnitude
Since pressure drop measurements are the primary
the conclusions are significantly affected by this uncertainty
above the Critical Micellar Concentration (CMC),
which generally still corresponds to a
motivation for this research, these measurements were
in channel dimensions.
fewThe
weight
Suchinlarge
are necessary
since the
concentration
using single-phase
flowbulk
of various
fluids through
device percents.
geometry is shown
figure concentrations
1. The flow of validated
glass capillaries.
Theseinterface,
validation tests
were
oil
sheared
off significantly
drops of water atwhen
a T-junction
as shown in begin
may
drop
the molecules
to cover the
given
theconducted
large
figure 1(a) [16]. The water drops are convected down the exactly like the actual experiments, except that the microfluidic
surface
tochannel
volume
ratios
length
of the
(figure
1(b)).involved.
A close examination of devices were replaced by round capillaries of known diameter.
For most practical purposes, simply adding a large concentration of a good sur1505
factant is sufficient for producing beautiful drops. In cases where things do not work
properly, one may try several different surfactant molecules and will generally find a
combination of fluids and surfactants that provide a satisfactory result, owing to the very
large number of available molecules1 . However, this large variety can be the source of
complications when the dynamic situations need to be understood, for example during
the creation of a new interface. Indeed, the dynamics of surface tension modification by a
1
This may be more or less difficult. For instance, there is a limited number of surfactants for fluorinated oils and their performance is not very good.
25
Surfactant
molecules
Drop
Figure 1.6: Variations of surfactant concentration due to surface dilation and contraction
as the drop shape varies. The width of the red region represents the local concentration.
surfactant is an active research subject for stationary interfaces [54, 49, 34]. When the effects of an unstationary flow are added, one quickly reaches many-dimensional parameter
spaces where little can be said that is of general use.
For instance, the adsorption/desorption dynamics can vary for different chemical
compositions, leading to faster or slower action of the surfactant molecules. Therefore,
the amount of surface coverage on the drop detachment shown in Fig. 1.3 may vary for
different chemical combinations. In some cases, Marangoni effects will play a role in the
drop breakup, while their effects may be negligible in other cases [69].
A more tractable difficulty arises when we allow the surface to dilate: Even though
the volume in the drop must remain constant for incompressible and insoluble fluids, a
change in the shape of the drop implies that certain regions will go through a surface
dilation while other areas will contract (Fig. 1.6). This effect will automatically lead to
variations in the surface concentration of surfactant which, in turn, will apply solutocapillary (Marangoni) stresses [50, 64]. This is shown schematically in Fig. 1.6 where we
show a circular drop, initially evenly covered with surfactant, going through a transition
to an ellipsoidal shape. The poles of the drop contract and therefore see their surface
concentration increase, while regions near the equator see the surface concentration decrease.
Situations in which drops go through large shape changes are very frequent in microfluidics, for example as drops flow through rapid contractions or expansions in the
channel geometry. In Fig. 1.7, the relaxation of the drop shape to a circle is used by
Hudson et al. [66] to measure the interfacial tension between the drop and the carrier
liquid. Although the authors do not address questions of Marangoni effect, it is likely
that it will introduce a measurement error in the presence of surface active molecules.
This effect was also observed by Bremond et al. [23] when drops were pulled into channel contractions. The authors there observed the creation of “nipples” which favored the
merging of drops, in part because those regions were impoverished in surfactant through
surface dilation.
Another important effect is due to the surfactant transport along the interface. For
instance, the molecules can be swept by the external flow along the drop surface and
accumulate at the stagnation points. In the reference frame of the moving drop, the
semi-parabolic flow profile in the external fluid must be equal to the drop velocity at
two points. This leads to two stagnation points which separate the flow into “external”
26
081905-2
Hudson et al.
Appl. Phys. L
FIG. 2. Freeze-frame image of drops flowing left to right in an extensional
flow gradient !water drops in pdms 1000". Measurements can be done at
Figure 1.7: Measurements
of surface
observation
the theshape relaxation of a
either the entrance
or exit of atension
constriction;by
the exit
is shown here. When
drops
leave
the
constriction
!the
channel
walls
appear
as
slanted
lines
in the
drop towards a circle [66].
FIG. 3. Experimental analysis of drop deformation
left half of the image", the flow decelerates in proportion to the change in
channel constriction !Taylor plot": %"c!5 / !2"ˆ + 3"
cross-sectional area. The drops, which are generated periodically in time,
of D / ao. The radius ao, deformation D and traject
therefore, become closer together. This deceleration corresponds to a
sured directly by image analysis, at a rate in exc
stretching in the transverse direction; note that the drops at the left side are
The slope is equal to the interfacial tension !mN/
stretched vertically, and their deformation decays as they pass to the right.
Here for illustrative purposes the drops are allowed to come relatively close
together !which is suboptimal for measurements, because eventually their
Anatech" and then sealed to glass. Finall
relaxation is hindered". Generally, the drop production rate is adjusted so
that there are a few drops per image.
treated to return the PDMS channels t
postulate a tempo
has yet to be con
Microfabricate
gular cross sectio
condition.
andwere
dyn
A variety oftopology
immiscible fluids
the velocity. Specifically, for flow along x, the extension rate
onstrate
a
capability
to
measure
a
wide
is written:
rectangular cross
tension !Table I". For each experiment,
−2
2
2
!˙ = du/dx = − !dt/dx" d t/dx ,
!2"
applications.
Fig.
recorded, so that
the Arrhenius-adjusted
used in data analysis.
where t is the time since the drop entered the field of view
two-dimensional
Flow was driven by computer contr
and u is drop velocity. When drops deform, D depends in
Figure 1.8: Flow
profile around stagnation points, proposed by Taylor
[118].New
through
the chann
!from either
Era or Harvard
Appa
general not only on !˙ but also on drop history. The response
the
microfluidic
device.
Flow
rates 50
we
dD / dt is proportional to the difference in D from the steady
labelled
with
16
drops
were
produced
at
a
frequency
som
value of D at the instantaneous value of !˙ :
channels,
theD exce
gas
camera
frame rate
and exhibited
and “internal” regions, as shown in Fig. 1.8. If the flow carries with it
molecules
that
5
dD
imaged
by
bright
field
optical
microsco
= Dsteady
− Dinterface,
=
#!˙ −those
D, will display strong
!3"
energetically prefer adsorbing
to the
adsorption atliquid
the segments a
dt*
using either a 4& or 10& objective len
2"ˆ + 3
stagnation points where the velocities are weak and the molecules spend
periods.
in
the2".channel
nearlonger
their edges
!Fig.
The image cor
acqu
*
is the
nondimensional
time
t / #, and by the flow, eventually finding
These molecules willwhere
thent be
swept
along the
interface
50 Hz, althoughtopology
higher rates have
b
of also
segm
sure
time
for
each
image
is
chosen
ˆ
ˆ
" + 3"!19" + 16"
"cao at%"the
themselves trapped at the!2stagnation
points
cao rear of the drop. This scenario is
#=
=
!4"
pa
300 's" that thegeometries.
motion of drops In
during
$ and the carrier fluid,
"ˆ + 1" of $
true regardless of the relative 40!
velocities
the drop
since the drop
a full pixel andbubble
therefore cap,
causes the
no blu
to
will always travel faster
the outer
very
close
to theaowalls.
thisorreason,
is the For!1.85
is thethan
characteristic
timefluid
for drop
shape
relaxation,
/ pixel,
0.738 'mthe
126respectively"
experime
Cerro
radius, "ˆ is the relative viscosity of the drop
diffraction grating.
Typical image
size is
front-rear symmetry undistorted
is alwaysdrop
broken.
The imagescapillaries
are analyzed inthat
real time
"d / "c, and % is a function of "ˆ . dD / dt is the material time
w
Since these molecules
tend to show a strong asymmetry betweenequal
adsorption
and
to the camera frame rate" to obtain
derivative of D, which in time-invariant flow is u ! D / !x. To
desorption, they are avoid
energetically
bound
on the
interface
we can
athe drops
Fig.
10 shows
andobserve
radius of
as they p
the assumption
that to
theremain
instantaneous
deformation
ap-and mation
17
age window.
Sinceare
each frame usually c
!i.e.,the
dDrear
/ dt # of
0", the
wedrop.
calculate
Eq.accumulation
!3"
Dsteady at
strong accumulation proximates
of surfactants
The
effects
flow through rec
thisconcentrations
system collects more than 100 data
i.e., %"c!5
/ !2"ˆ +the
3"!˙ −surfactant
u!D / !x" vs Dis/ apresent
o !which we
especially important directly,
in situations
where
in weak
Periodically !e.g.,
every 2 s", in
droppart
trans
velocities,
name a Taylor plot", so that the slope is $ !Fig. 3".
because strong surfactant
gradients can build up, which lead to drops
slowing down
formation are fit to polynomials !the resu
Although the velocity field in a channel can be computed
The liquid film of
significantly with respect
to geometry
the external
flow. the
Indeed,
the superposition
of the Marangoni
polynomial
order ranging from 5 to 7" to
from its
!by solving
Navier–Stokes
equation"
that
du /external
dx = !˙ . At the
same are
time, relevant,
%"c!5 / !2"ˆ +
we are
concerned
here only
withmust
the velocity
of the drops
and externally imposed
flows
yields drops
that
travel slower
than the
phase,
themselves.
Therefore,
our
analysis
needs
neither
the
deto
a
linear
function
of
D
/
a
o !Fig. 3", to
if the Marangoni flow acts on the whole drop. These aspects have not yet been studied
thereby reduces
th
tailed geometry nor the entire flow field. Instead, we measure
Measurements of $ ranging from
in detail in the literature.
our work,
weeliminating
have addressed
relatedhave
issues,
we As
slugs.
a I".
result
the dropInvelocity
directly,
the need some
to calibrate
beenwhich
demonstrated
!Table
Sma
discuss in Section 4.1.
each device.
always be studied by decreasing the flow
moves across th
deviceofissurfactant
fabricated byisa rapid
protohand, the
A third issue dueThe
to microfluidic
the presence
the introduction
of maximum
a surfacemeasurable value of
typing method, using a thiolene frontal photopolymerization
corners
are and
alway
the camera frame
rate, viscosity,
a %
viscosity term. This !FPP"
effecttechnique
is well known
the18studies
of foam
drainage
[100] but has
not/ w ", where w / wo i
or conventional
SU8
phoreported in
earlier
proportional
to 1n!w
c
c
and nanoparticles
been addressed in thetolithography.
microfluidics
community
specifically.
Other
exotictiotwo-dimensional
Typical
cross section
of the deformation
chanof the channel constriction %which in
nel ranged from 90& 700 'm to 850& 1250 'm!height h
gas–liquid–
Design tradeoffsalong
for various
measuremen
& width w". The width wc of the constriction is typically
Here, use of a considerable range of v
film and menisci
27
w / 3. Transparent-elastomer !polydimethylsiloxane, PDMS"
drop viscosity is reported. Although the
replicas are treated with low-power oxygen plasma !SP100,
$ are insensitivedeposition
to drop size ao,patter
its contr
Downloaded 21 Nov 2006 to 129.104.38.4. Redistribution subject to AIP license or copyright, see http://apl.aip.o
nanoparticles in
electron microg
(Fig. 10e) after
phase transitions can also appear through the accumulation of molecules on the interface,
leading to variations in the mobility of the interface [94, 29]. While all of these questions
are fascinating, they have not yet been studied and will not be addressed further in this
document.
Finally, an effect which we have found to be of major importance has been the effect
of heating on surfactant repartition. We have observed anomalous Marangoni flows, i.e.
going from the cold to the hot regions, when heating a water-oil interface with a focused
laser. The direction of these flows implies that the surface tension has risen due to heating,
which is contrary to the traditional statistical mechanical view of interfaces. These aspects
will be developed in Chapter 3, where we also show measurements of surfactant transport
due to the heating.
1.2.3
Singular events: Formation, merging and splitting
The most violent dynamics happen during singular events, namely those involving breaking or merging of interfaces. The dynamics of these events happen on very short time
scales, since they are associated with finite-time singularities, in the case of breaking, and
with infinite curvatures at the two touching interfaces, in the case of merging.
During the breakup process, a particularity of microfluidic devices is their confinement, since confined two-dimensional threads cannot break into drops, for example in a
Hele-Shaw cell [36]. Therefore, the breakup mechanism that leads to the drop formation
can only be based on the Plateau-Rayleigh instability at the very late times, i.e. times
when the width of the thread becomes smaller than the channel height.
A geometry for producing drops that has been studied extensively is the flow focusing geometry, shown in Fig. 1.9. Since the initial work of Anna et al. [7], several groups
have studied in the detail the breakup process of a water or air thread that is confined
in a flow-focusing region [58, 45]. It has appeared that the breakup is largely dominated
by the imposed flowrate of the external phase, the Plateau-Rayleigh instability only appearing at the very late stages. On a related topic, Guillot et al. [60] showed that the
confinement of a thread of water in oil could also lead to the complete suppression of the
breakup of this thread in a microchannel. The particular geometry of the microchannels
therefore plays a major role in the formation of drops, on their size and size dispersion,
as well as on the precise location of breakup.
Merging events are different in that they are mainly due to interactions between
droplets and are less dependent, a priori, on the geometry. Particularly, the presence of
surfactants in the solution is known to stabilize emulsions [18] through several mechanisms, namely through preventing the drainage of the lubrication film that is trapped
between two adjacent drops. Significant work has focused on producing droplet merging
by manipulating the geometry of the channel and thus manipulating the distance between drops [113], as well as their shapes [23]. However, there does not yet exist a clear
picture regarding the mechanisms that passively force droplet merging, although active
techniques have been explored, as mentioned below.
28
Appl. Phys. Lett., Vol. 82, No. 3, 20 January 2003
FIG. 2. Experimental images of drop breakup sequences occurring inside
the flow-focusing orifice. (a) Uniform-sized drops are formed without visible satellites; breakup occurs inside the orifice. The time interval between
images is 1000 $s; Q o !8.3"10#5 mL/s and Q i /Q o !1/4. (b) A small
satellite accompanies each large drop; breakup occurs at two corresponding
locations inside the orifice. The time interval between images is 166 $s;
Q o !4.2"10#4 mL/s and Q i /Q o !1/40.
through a small orifice that is located downstream of the
three channels. The outer fluid exerts pressure and viscous
stresses that force the inner fluid into a narrow thread, which
then breaks inside or downstream of the orifice. In the experiments reported here, the inner fluid is distilled water and
the outer fluid is silicone oil !viscosity, 6 mPa s", which leads
to water drops that form in a continuous phase of oil. Span
80 surfactant !Sorbitan monooleate, Aldrich" is dissolved in
the oil phase at 0.67 wt %. The surfactant solution was prepared by mechanically mixing the two components for ap-
‘‘bottom-up’’ aphe level of indieal for thinking
mation.6 For exmicrofluidic deater stream at a
aried in size demilar microfluidic
flows have been
añán-Calvo and
as bubbles, less
e called capillary
y tube into a bath
orifice, and the
external liquid
thin jet, which
es. In a separate
duce liquid drophis flow-focusing
pstream capillary
FIG. 1. Flow-focusing geometry implemented in a microfluidic device. An
orifice is1.9:
placedA
a distance
H f !161 % m downstream
of three
Figure
“flow-focusing”
geometry
is coaxial
used inlet
to produce water drops in oil. The right
streams. Water flows in the central channel, W i !197 % m, and oil flows in
columns
show
different
that
the two outer
channels,
W o !278 regimes
% m. The total
widthcan
of theappear
channel is for different forcing conditions. From
W!963 % m and the width of the orifice is D!43.5 % m. The thickness of
Ref.
[7].
the internal walls in the device is 105 %m; this thickness is necessary in
nts using a floworder to obtain a uniform seal between the glass cover slip and the poly!dimethylsiloxane" !PDMS". The uniform depth of the channels is 117 %m.
The ‘‘design’’ dimensions were slightly different than the measured values
reported here since silicone oil swells the PDMS.
Manipulating drops
proximately 30 min and then filtering to eliminate aggregates
and prevent clogging of the microchannel.
The fluids are introduced into the microchannel through
flexible tubing and the flow rate is controlled using separate
syringe pumps for each fluid. In the experiments reported
here, the flow rate of the outer fluid !oil", Q o , is always
greater than the flow rate of the inner fluid !water", Q i .
Three different flow rate ratios are chosen, Q i /Q o !1/4,
1/40, and 1/400, where the oil flow rate given corresponds to
the total flow rate for both oil inlet streams. For each Q i /Q o ,
oil flow rates spanning more than two orders of magnitude
are chosen (4.2"10#5 mL/s#Q o #8.3"10#3 mL/s). At
each value of Q o and Q i , drop formation is visualized using
an inverted microscope and a high-speed camera.
In Fig. 2!a" we show formation of a nearly monodisperse
suspension of water droplets with diameter comparable to
the orifice width. Breakup occurs within the orifice. In addition, drops may break within the orifice such that one or
more satellite droplets are formed in a regular and reproducible manner. Figure 2!b" illustrates this kind of breakup process, which thus naturally forms a bidisperse suspension.
We have conducted many experiments documenting the
formation of two-phase liquid–liquid dispersions in microchannels fabricated with the flow-focusing configuration. A
phase diagram indicating the range of responses we have
observed is shown in Fig. 3. We observe the formation of
1.3
In parallel to all of the fundamentals
above, the most important
364
© 2003 American detailed
Institute of Physics
distribution subject to
AIP license
or copyright,drops
see http://apl.aip.org/apl/copyright.jsp
how
to manipulate
in the microchannels in order to perform the
1.3.1
Channel geometry
Anna, Bontoux, and Stone
question remains
necessary operations in a Lab-on-a-Chip (LOC). In a short review article, Joanicot and Ajdari [70] listed
a few operations that were necessary in order for droplet microfluidics to live up to its
promise. They list fabrication, sorting, storage, fusion, breakup, and trafficking as particular operations that must be performed on drops. They also distinguish, with reason,
active from passive control, as well as distinguishing external controls from integrated
droplet-level controls. Below, we will place in context the different control mechanisms,
from the most basic to the most recent. We begin in this section with the two traditional
approaches that have been used to provide some control: The microchannel geometry
and the external flow variations, e.g. through syringe pumps.
29
365
The channel geometry is the single most important element of the microfluidic control
of droplets and this fact, though trivial, should always be kept in mind when designing
control mechanisms. Very early on, it became clear that the geometry of the channel
could be used to provide many actuation operations. We have already seen above how a
zig-zag shaped channel can be used to mix the contents of a drop without any external
interference [107] and how droplet formation depended on the geometry in which the
drops are formed [46, 7]. Moreover, successive bifurcations were used to break initially
large drops into a dense and monodisperse emulsion, as shown in Fig. 1.10 [82].
In these cases, the robustness of this control approach is due to the system functioning in a stable regime, meaning that a small uncertainty in the fabrication or the external
control only leads to a small uncertainty in the result. Since nonlinearities are introduced
by the presence of drops, for example during the pinch-off of a new drop [45], those can
lead to unstable regimes in which the desired flow is not obtained. For instance, the
breakup of drops in Fig. 1.10 may become unstable and the liquid may flow through only
some branches, for certain operating conditions. This would be the case if the resistance
when a drop enters the channel is decreased compared to the resistance of the channel
measure when the drops are collected.
number separating regions of
In some applications, the T-junction design may reg drops in Fig. 2(k) can be
quire too much space on the device or applied pressures
how droplets are stretched in
tch rate G. Here we define an
number CG ! !Ga=", where
ed radius of the drop. For an
eighborhood of a T junction
idth w0 , the stretch rate is G "
xtensional stresses at the stagmuch larger than the shear
nction, change the drop length
e we are seeking the critical
op just breaks, we write ‘e #
G0 is the upstream shear rate
FIG.of3.bifurcations
Sequential can
application
passive divide
breakup.
In into a
Figure
A cascade
be used toof passively
droplets
n with G0 & G,
the 1.10:
capillary
(a)
drops
are
formed
at
high
dispersed
phase
volume
fractions
emulsion. From Ref. [82].
CG / $‘e =a #monodisperse
‘0 =a%2 . We take
(drop formation not shown) and are sequentially broken to form
op is at the stability limit, and
small drops. In (b) the droplets flow downstream with nearly
single
phase defect-free
[14, 51]. We
have explored this ordering.
possibility, which is described in
vation, ‘0 w20 !with
‘e w2ea, to
obtain
hexagonal-close-packed
Section 2.1.
054503-3
The destabilizing effects can however be corrected by robust design of the geometry, for example by shunting two channels together at a location which depends on the
distance between drops, in order to equilibrate the pressures in the two branches after
the bifurcation [40].
Figure 1.11: A spiral channel is used to impose both elongation and shear on flowing
drops. See Section 4.1 for details.
Since the micro-fabrication methods allow any two-dimensional shape to be made,
more subtle uses of the channel geometry to control the flow of drops has also been
shown, for example in order to increase the space separating the drops or to move them
closer together. This may be achieved by reducing or increasing the channel width. In
Section 4.1, we will describe the use of circular shaped channels to submit drops to a
constant shear, which can be used to elongate drops in a similar fashion to Taylor’s four
roll mill [117]. Figure 1.11 shows an early version of the experiment where a logarithmic
spiral microchannel is used to produce progressively increasing shear and elongation on
drops.
30
1.3.2
Imposed external flow
Along with the channel geometry, the second most important method for controlling
droplet operations is by externally controlling the flow. Since most two phase flows
in microchannels are produced either with positive displacement pumps (e.g. syringe
pumps) or through a controlled pressure source, the magnitude of the forcing is a natural
and powerful way to control the way drops are formed, broken, or transported. For
instance, the size of drops that are produced in the flow-focusing junction of Fig 1.9 is
mainly controlled by the relative flowrates of oil and water, in addition to the geometry
of the focusing region. In this way, increasing the water flowrate will increase the size
of the drops by “inflating” the drop more during the time that the neck is pinched off.
Alternatively, varying the oil flowrate changes the time available for the water to flow
and by the same token changes the size of drops thus produced.
!"#$%&
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!"#$%&
%'()*+,-./)-#"0-1/2./3*/4/5/6,70"8/28(/39/4/:$;//6,70"8$
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Figure 1.12: A snapshot showing
water in oil drops being produced through a flow fo&'()*+,-./)-#"0-1/2./3*/4//5:/6,70"8/28(/39/4/5/6,70"8$
cusing device showing one drop per period (top) and nine drops per period (bottom).
Experimental conditions for the top image: Qoil = 2 µL/min and Qwater = 0.5 µL/min.
Conditions for the bottom image: Qoil = 20 µL/min and Qwater = 2 µL/min.
!"#$/5:
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While this approach is very useful in practice, several issues must be kept in mind.
The first is that changing the flowrates not only changes the drop size, it also changes
the frequency at which they are produced and the velocity at which they travel. One
would ideally like to control these parameters independently, particularly when the drop
production is coupled with a complex microfluidic device downstream. Furthermore,
variations in the flowrates can also produce changes in the flow regime, not only in the
size of the drops. Some early measurements, made in our lab, showed that the number of
drops produced at each breakup event could be varied by changing the flowrates of the
external and internal fluids (oil and water, respectively), as shown in Fig. 1.12.
1.4
Active
integrated manipulation
5<///
For droplets to be truly viable as Lab on a chip (LOC) vessels, global control through the
means presented above will probably not be sufficient. For example, intelligent operations
need to be performed
based on the drop size, its contents, or other measurable quantities.
5<///
31
These operations can consist of sorting, merging, or dividing the drops asymmetrically. To
this end, several approaches have been explored, involving different physical mechanisms.
1.4.1
Drops on an open substrate
Annu. Rev. Fluid. Mech. 2005.37:425-455. Downloaded from arjournals.annualreviews.org
by Pr. Jean-Marc Chomaz on 03/18/05. For personal use only.
It is important to take a small detour here to note that a second approach, parallel to the
type of microfluidics we are talking about, has been developed for manipulating drops on a
flat substrate or in suspension in a quiescent fluid. Two major categories of manipulation
have taken shape. The first and most widely studied is based on using electrical forces to
induce a force on10the
drop14:30
through
dielectrophoresis,
or electro-wetting
P1: IBD
Nov 2004
AR electrophoresis,
AR235-FL37-16.tex AR235-FL37-16.sgm
LaTeX2e(2002/01/18)
(see e.g. [95, 97, 119]). The second approach is to modulate the surface tension of the
drop, through heat or chemistry, in order to produce an imbalance in surface stresses
which in turn lead to motion [101, 42, 74, 41]. Other approaches were also explored,
including the use of Surface Acoustic Waves (SAW) [126, 98], acoustic streaming [86], or
443
MICROFLUIDIC ACTUATION BY SURFACE STRESSES
optical tweezers [112, 27].
Figure 7
(a–e) Image sequence showing the thermocapillary splitting of a dodecane droplet
Figure 1.13: A microfabricated
array
of electrodes
is used
tototalmanipulate
dropsapplied
on a tosolid
= 8.5 s). The voltage
the
on a partially
wetting
stripe (w = 1000
µm, !t
embedded
microheaters
(electrical
resistance
155
")
was
2.5
V
(Darhuber
et
2003b).
substrate, from Ref. [42]. Drops are shown to move in a straight line, turn, and al.split
(f–i) Image sequence of dodecane droplet propelled by thermocapillary forces. Intersecting
orange stripes designate partially wetting regions (w = 1000 µm, !ttotal = 104 s) (Darhuber
Different operationsethave
been implemented regarding drops on a substrate, includal. 2003b). (j–l) Dodecane droplet turning a 90◦ corner (!ttotal = 164 s) (Darhuber et al.
ing forming, moving, dividing,
and Evolution
fusing drops,
asinstability
shown atintheFig.
1.13,
drops
are
2003b). (m–p)
of fingering
leading
edgewhere
of a silicone
oil film
advancing
on
a
chemically
homogeneous
substrate
(silicon
wafer)
due
to
thermocapillary
manipulated using a set of patterned electrodes on the surface [42]. In this approach to
forces (applied shear stress τ = 0.18 Pa, !ttotal = 17 min, instability wavelength λ ≈ 500 µm)
manipulating droplets, the
fundamental advantage is that each drop can be controlled
(Cazabat et al. 1990).
at every moment, anywhere in the system. However, one loses in the process the helpful
effect of geometry in applying2004;
standard
operations
on &drops
which
the
Darhuber et
al. 2003b,c; Ford
Nadim passively,
1994; Smith 1995;
Yarinmakes
et al. 2002).
manipulation much slower and
more(1989)
complex
Furthermore,
Brochard
studiedtotheimplement.
migration of 2D
droplets driven bythese
a smallsystems
thermal
similar force
balancewith
as usedfrom
for chemically
graduated
surfaces
(secalso suffer from several issuesgradient.
whichA must
be dealt
a practical
point
of view
tion 4.1) was used to describe droplet motion subject to spatial variations in γlv , γls ,
such as sensitivity to surfaceand
contamination,
contact angle hysteresis, and evaporation.
γsv . Depending on the relative strength of the driving forces at each interface,
Finally, it is not clear that these
systems
willtoward
alloworthe
of ofvolumes
as small
the droplet
can move
awaymanipulation
from the cooler end
the substrate.
When
contact
angle
hysteresis
is
explicitly
incorporated
into
the
model
(Brzoska
et al.
as those available for in-channel manipulation, to which we now switch.
1993), droplet motion is only possible above a critical droplet radius (neglecting
thermal variations in γsv and γls ) for a given thermocapillary stress. Above this
critical value, the flow speed increases linearly with the droplet radius R and ap1.4.2 Drops in a microchannel
plied thermal gradient ∇T . These predictions were confirmed by experiments with
silicone oil droplets
(2 !also
R ! 10been
mm) moving
hexadecyltrichlorosilane-coated
Some of the same physical mechanisms
have
used on
toa manipulate
drops inside
silicon wafer (11 ! θs ! 13◦ ) subject to |∇T | < 1◦ C/mm. In their model Ford &
microchannels. Here, the focus
has(1994)
not been
a the
drop
at any
location
and
Nadim
relievedon
themanipulating
stress singularity at
contact
line by
incorporating
a Navier slip condition and examined larger thermal gradients by allowing surface deformation beyond
32 a cylindrical cap. The droplet speed then depends on the
Navier slip coefficient, which is exceedingly difficult to measure experimentally.
Smith (1995) conducted a full hydrodynamic analysis in the lubrication limit for
the requisite encapsutric field, all drops flow into the waste channel which is
ening requires a much
shorter, and thus offers lower hydrodynamic resistance than
to individual microrethe second, collect channel. To direct drops into the collect
is best achieved using
stream, we energize electrodes under the channel in the sortbles formation of uninipulation,4 complex
l volumes.1 However,
crofluidic device is esal library of 108 – 109
1 kHz to be practical.
or particles and cells in
osmotic,6 mechanical,7
and dielectrophoretic
drops have been made
FIG. 1. !a" Schematic top view of the device. Water drops formed by flow
limited to 10 Hz Ref.
Figure
1.14:
Drops
flowingphase
in an
oil flow
flowinto
canthebewaste
sorted
either
the right or left
focusingof
in water
the continuous
of oil
channel
sincetothe
ng charge, but this reresistance
of the
waste channel is smaller
than
that of the collect channel. !b"
10,13
outlet
channels
through
dielectrophoretic
forces
[5].
within the drops.
Schematic cross section of the device. The molded PDMS microfluidic
priate for drop sorting
channel is aligned to the 30-!m PDMS layer which is spin coated on the
tic sorting canatachieve
ITO electrodes.
!c" Ininstead
the absence
an electric
field, water drops
any time; patterned
the approach
has been
to of
allow
the geometry
to apply the large
flow
into
the
waste
channel.
!d"
Applying
an
electric
field,
the
drops
are the dynamic
eling. Thus, improved
forces on the drops, passively, and augmenting this passive forcing with
attracted toward the energized electrode and flow into the collect channel.
actuation at Transparent
particular ITO
locations
in the microchannel. In this fashion, a train of drops
electrodes have been drawn in gray for grounded eleccan flow passively
through
theenergized
networkelectrodes.
of microchannels and intermittent forcing can be
trodes and
white for
used when particular operations are necessary.
Of 024104-1
the operations that were listed by
Joanicot
and Institute
Ajdari, of
only
sorting and fusion
88,
© 2006
American
Physics
.5. Redistribution
subject
to AIP license
or copyright,
seeboth
http://apl.aip.org/apl/copyright.jsp
had
been published
by the
end of 2006,
through the application of electrical forcing.
In the first case, dielectrophoretic forces were shown to be sufficient to sort drops a few
tens of microns in size, as shown in Fig. 1.14 [5]. Here, a 1kV power source was used to
produce forces in the range of 10 nN, yielding a transverse velocity of about 1 cm/s for
a drop 12 µm in diameter [5].
More recent implementations of this technique have shown that it is very easy
to include the electrodes along with the fluidic channels, by fabricating a network of
“channels” for the electrodes, in parallel to the fluidics, and filling them with a low melting
point metal or conducting plastic. In this way, simply connecting the electrodes to an
AC power source is sufficient for producing forces on drops. A second advantage of this
technique is its rapid reaction time, shorter than 1 ms (the calculation in Ref. [5] is
incorrect and underestimates the time to produce the force to a few µs). Note however
that the dielectrophoretic force applied on a drop is a body force which decreases as R3 ,
as the drop radius R decreases.
The second operation that was demonstrated by 2006 was the fusion of drops [33,
4, 96]. The barrier to fusion is generally in the surfactant coverage of the interfaces which
prevent the drainage of the lubrication film that separates the two drops. Priest et al.
demonstrated that the “electrocoalescence” was due to a hydrodynamic instability of the
interface, generated by the imposed electrical field. They also showed that two drops
could be merged using a few volts and with a good selectivity even in a crowded flow, as
shown in Fig. 1.15.
The combination of the two above operations with geometric forcing was sufficient
for Raindance Technology, a startup spun out from the Weitz group at Harvard University,
to begin its operations (http://www.raindancetechnology.com/). Although Raindance
probably has a more complete tool set for manipulating drops, those tools have not been
published. The remaining operations, namely the control of fabrication, storage, breakup
or trafficking, remained unavailable until recently.
33
Figure 1.15: Electrocoalescence of droplet pairs in a crowded flow, from Ref. [96].
In this context, two papers originating from our lab were published in 2007 [12, 11]
and provided an approach to control the formation and division, in addition to sorting,
synchronizing, and fusing drops. The technique is based on inducing a Marangoni flow on
the interface, whose net effect is to produce a force pushing the drop. More recently, we
have also demonstrated more complex operations such as storing drops, reordering them,
and trafficking them in a three-way exit [37]. This technique will be treated in detail in
Chapter 3.
1.5
Manuscript organization
The rest of this document is organized as follows: We begin in the next chapter with
our work on multiphase flows in complex geometries, treating the problems that we have
looked at in chronological order. This is followed by two of the articles we have published
on the subject. Chapter 3 then describes the main results on the technology we have
developed for manipulating drops using a focused laser, also followed by two reprints on
the subject. Finally, some more recent and unpublished work concerning the flow of drops
in microchannels is presented.
34
Chapter 2
Drops in funny channels
35
Networks of channels are omnipresent in nature. When they are present in animals and plants, they serve the clear purpose of transporting matter (e.g. nutrients or
oxygen) to and from organs, as shown for example in the networks of recent interest of
Fig. 2.1. These naturally occurring networks can display a very wide range of sizes and
can involve different physical interactions between the flowing fluid and the network walls.
Furthermore, networks have different topologies and follow different scaling laws; while
leaf venation forms a redundant network allowing several routes between two points, the
tree itself provides no loops and no redundancies to reach a particular branch.
Figure 2.1: Four networks which transport nutrients: The blood circulation in the
retina[1], the morphogenesis of the chicken circulatory network [120], a walnut tree
[Matthieu Rodriguez, LadHyX], and a venation network in a tree leaf [20].
In other cases, such as for the flow in porous materials, the network representation
is a simplification of a structure that may be too complicated to be generalized (see
e.g. [26, 109]). In this case, the scientist’s job is to produce a simplified representation
that accounts for the transport processes in the porous medium while using a geometry
that can be described quantitatively. Microfabrication techniques were introduced as an
experimental tool to produce some model porous materials by Lenormand in the early
80’s [80], as shown in Fig. 2.2, but have not been explored very extensively.
The network that initially motivated our work is the human lung’s airway tree,
generally modeled as a branching binary tree of circular tubes, as shown schematically
in Fig. 2.3. On close inspection, the lung represents a fascinating object for a fluid
dynamicist; it begins at the trachea whose diameter is about two cm, where flows reach
36
Figure 2.2: Liquid-gas menisci in a microfabricated model porous network, from Ref. [79].
Reynolds numbers of a few thousands and where the flow is typically turbulent. After
a succession of bifurcations, the scale of each channel reaches a few millimeters (around
generation 5), corresponding to low Re flows. The sizes decrease further until they reach
a few hundred microns at the alveoli where the transport of oxygen becomes dominated
by diffusion in a rarefied medium rather than by hydrodynamics [122].
Figure 2.3: Classical image of the successive bifurcations in the human airway tree, from
Ref. [122]
The story however is much more complex due to other interactions, of which we
will note two: (i) The lung also undergoes large deformations during respiration and
the flow interacts with the elasticity of the supporting structure. The deformability of
the structures and the mechanical characteristics therefore have a major influence on the
37
distribution of the flow in the branched network. (ii) The airway walls are constantly
lined with a liquid layer which plays important physiological roles, such as extracting
inhaled contaminants or reducing the surface tension of the alveoli in order to prevent
lung collapse. The liquid lining can interact with the flowing air in several ways, namely
by forming liquid plugs that block the gas flow in the airways during disease.
Below, we will consider experimental models that explore multiphase flows in branching channels and which may involve interactions with elasticity. In doing so, we will also
address some questions of interest in microfluidics regarding transport of drops and bubbles.
2.1
Flow of a low viscosity finger through a fluidfilled bifurcation
The results of this section were published in collaboration with Sedina Tsikata
and Matthias Heil [14].
The first question we will ask is to know how a low viscosity fluid penetrates in a
branched structure that is filled with a high viscosity immiscible fluid. The question of
replacing one fluid with another of lower-viscosity is of obvious importance in the case
of oil recovery, where high viscosity oil must be pumped out through a porous medium
by water. In the case of physiological flows, this situation arises in the case of the first
breath a baby takes at birth, where the liquid initially filling in the lung must be replaced
with lower viscosity air. A similar situation also arises in the case of an air bubble that
flows in the blood stream, for example as a treatment for cancerous regions [30].
A reduction of this problem to a tractable form can be made by considering the
splitting of a low-viscosity finger, pushed at constant flowrate Q through a bifurcating
channel. The situation is shown schematically in Fig. 2.4, along with the definition of
terms used below. The question that will determine the filling of the network can be
stated at the level of one bifurcation by asking whether the low viscosity finger will
divide symmetrically or asymmetrically.
2.1.1
Linear stability analysis
A linear stability analysis can be made to determine the stability of the symmetric solution
to small perturbations. In the case when the two daughter branches are open to the same
exit pressure (e.g. atmospheric pressure), the stability arguments are similar to those
used to justify the single finger selection in the Saffman-Taylor instability: Suppose that
the fingers initially branch symmetrically but that the velocity of the finger in branch
1 increases slightly. This is equivalent to saying that the relative length of viscous fluid
ahead of this finger decreases (L1 < L2 ), which in turn leads to the velocity difference
increasing further, since finger 1 feels less resistance than finger 2. This is an unstable
situation which implies that fingers branching into channels open to the atmosphere will
always branch asymmetrically. If this bifurcation is one link in a binary network, the low
viscosity finger will only open one path through the network, leaving most of the channels
full of viscous fluid.
38
L1 (t)
V1 (t)
)
U1 (t
Q
pdr (t)
U2 (t
)
L2 (t)
V2 (t)
Figure 2.4: A low viscosity finger is pushed at constant flowrate Q and penetrates in a
bifurcation filled with a high viscosity immiscible fluid. L1,2 denote the length of viscous
fluid ahead of the finger, and U1,2 the velocities in each branch.
In the case of pulmonary flows, one expects that some mechanism might stabilize the
symmetric branching pattern so we consider the role of the lung’s elasticity, by lumping
the elastic effects into a compliant chamber at the downstream end of each of the daughter
branches. The linear stability analysis should now be done carefully. We sketch the main
steps below, the details being available in Ref. [14] at the end of this chapter.
In each of the daughter tubes, the pressure drop across the occluded section has
three components: (i) the Poiseuille pressure drop ahead of the finger tip, ∆ppois =
µRLi (t)Qi (t), where the flow resistance R depends on the tube’s cross-section [123];
(ii) the pressure drop across the curved tip of the finger, ∆ptip = C(Ui ), where the
function C(Ui ) accounts for the radius of curvature which depends on the velocity of the
finger [24, 62]; (iii) the pressure pelast in the chambers at the end of the daughter tubes.
Provided the membranes that close the (otherwise rigid) chambers are elastic, pelast
depends
only on the chambers’ instantaneous volume and we assume that pelast i (t) =
Rt
k 0 Qi (τ )dτ + p0i , where k is an elastic constant and p0i is the pressure in chamber i at
time t = 0.
Assuming that the viscous pressure drop in the fingers can be neglected, the fingers
in both daughter tubes are subject to the same driving pressure pdr (t), and we have
Z t
Qi (τ )dτ + p0i for
i = 1, 2,
(2.1)
pdr (t) = C(Ui ) + µRLi (t)Qi (t) + k
0
where Qi (t) ' A Ui (t), the approximation sign accounting for the fact that there is a
liquid film left on the surfaces.
The linear stability analysis can be performed by first differentiating Eq. (2.1) with
respect to time and expanding it in terms of the driving pressure. The zero-th order
term simply states the conditions for the symmetric solution, for which we can write
Li (t) = l0 − Ut, where U represents the symmetric solution velocity.
We determine the stability of this solution by writing the velocities as Ui (t) =
bi (t), where 1, with similar expansion for all other quantities. We further
U + U
simplify the problem by assuming that the capillary jump at the interfaces will not affect
the stability, which allows us to write the condition for stability
39
ch
ran
1
b
Q
bra
nch
2
400 microns
Figure 2.5: Experimental image of the splitting finger at a bifurcation.
Z
bi (t)
U
bi (t=0)
U
du
=
u
t
Z
0
2U − k/(µR)
dτ.
l0 − Uτ
(2.2)
bi decay,
If the right hand side of (2.2) is negative, the perturbation velocities U
indicating that the symmetrically branching solution is stable. The denominator of the
integrand on the right hand side represents the instantaneous length of the fluid-filled part
of the daughter tubes and is therefore always positive. Perturbations to the symmetrically
branching finger will therefore grow if U exceeds the critical value Uc = k/(2µR). Hence
at small velocities, when the flow resistance is dominated by the vessel stiffness, the finger
will branch symmetrically. Conversely, at large finger velocities, the resistance to the flow
is dominated by the viscous losses and the propagating finger will tend to open a single
path through the branching network.
2.1.2
Experimental realization
Experiments were conducted in PDMS microchannels by bonding thin PDMS membranes
on top of the exit holes. The channels were initially filled with silicone oil v100 (µ = 0.1 Pa
s) and fingers of perfluorodecalin (PFD, viscosity µpfd ' 5 · 10−3 Pa s) were injected
towards the bifurcation at constant flowrate by using a syringe pump. The finger tip
positions were located on microscopy images, as shown in Fig. 2.5, and the velocities in
each branch were calculated from successive images. In this way, the difference between
the velocities of the two fingers were measured as a function of time.
Figure 2.6 shows the evolution of the finger velocity difference as a function of time,
for different flowrates Q. The experimental data display a non-zero velocity difference
for all flow rates, indicating that the symmetric solution does not necessarily correspond
to a zero velocity difference. However, fingers forced at low flowrate show a constant
velocity difference while (U1 − U2 ) continuously grows for fingers forced at high flowrate,
indicating a transition from symmetric to asymmetric splitting.
40
Total velocity (µm/s)
U1!U2 (µm/s)
450
300
61.78
107.3
125.0
252.6
478.8
606.7
767.8
941.6
1881
Asymmetric
150
Symmetric
0
0
0.2
0.4
0.6
Normalized time (t U / 2Lc)
0.8
1
Figure 2.6: Velocity difference in two daughter branches as a function of time. U1 − U2
is found to continuously increase for U > 479 µm/s, in agreement with the theoretical
prediction.
The agreement between experiment and theory is verified by calculating the critical
value of the capillary number, using thin shell theory for the chamber stiffness k. Using physical values estimated from the experiments, we calculate the predicted critical
capillary number at Cac = 5 × 10−4 , which compares favorably with the experimental
transition which was observed in the range 4.78 × 10−4 < Ca < 6.06 × 10−4 .
2.2
Flow of a plug in a straight channel
The work presented in Sections 2.2 and 2.3 was published in collaboration
with Cédric Ody and Emmanuel de Langre [91].
A more common occurrence in pulmonary pathologies is the presence of plugs of liquid that can occlude an airway locally. These plugs can form through the destabilization
of the lung’s liquid lining through an process similar to the Plateau-Rayleigh instability [47]. They are especially observed in diseases such as asthma, when the diameter of
the channels is reduced due to inflammation, or bronchiolitis in children’s lungs. Plugs
can also be injected into the lung in order to deliver treatment to the distal alveoles, as
in the case of Surfactant Replacement Therapy (SRT). During this procedure, surfactant
is injected as a plug into a premature baby’s lungs in order to replace the molecules that
have not formed naturally [52, 31], with the aim to deliver the liquid as far down in the
airway tree as possible.
In parallel, many questions remain on the transport of multiphase flows in microchannels, as discussed in Chapter 1. We therefore set up an experimental study to
understand the relationship between the driving pressure and the velocity at which a plug
of length L0 travels in a straight microchannel with rectangular cross-section.
The experimental setup consisted of a Y-shaped junction in which plugs of PFD
were formed, as shown in Fig. 2.7, connected to a straight channel. The contact angle
41
between the PFD and the PDMS is small and we consider that the fluid is perfectly
wetting in our analysis. The size and position of the plugs as a function of time were
measured for different driving conditions, using high-speed video microscopy.
150µm
AIR
Liquid plug
Length
Liquid (PFD)
Figure 2.7: Formation of a plug in a microchannel. Air is forced at a constant pressure
while a controlled length of liquid is injected from a syringe pump.
2.2.1
Model
Given a constant driving pressure Pdr , the steady-state pressure balance across the plug
is
r
a
Pdr = ∆Pcap
+ Pvisc + ∆Pcap
,
(2.3)
r
a
where ∆Pcap
and ∆Pcap
express the capillary pressure drops at the receding and advancing
interfaces of the plug, respectively, while Pvisc represents the viscous dissipation occurring
150µm
in the bulk. Using Poiseuille’s law,AIR
the latter is expressed
as
Pvisc = αµL
0 Uplug
0,
Liquid
(2.4)
with α a dimensional coefficient corresponding to the geometry of the channel. For a
rectangular geometry [110], α may be approximated as
Length
−1
12
25 b
α' 2 1−6 5
.
b
π w
(2.5)
Liquid (PFD)
eb (ewi )
θa
b (wi )
L0
U0
1
Figure 2.8:
Figure 1: Sketch of the plug (sections A and B). Here, θa is the dynamic contact
Sketch
pluginterface
propagating
leftthickness
to right
in adeposited
two-dimensional
angle atof
theafront
and eb (efrom
of the
film at
wi ) is the
the rear of the plug in the thickness (width).
42
!3
x 10
2.5
2
1.5
channel.
r
a
When the plug is at rest, static values of ∆Pcap
and ∆Pcap
are given by Laplace’s
r
a
−1
−1
law as ∆Pcap = −∆Pcap = γκ, where κ ' 2 (b + w ) is the mean curvature of each
interface at rest. When the plug is moving, the front and rear curvatures are modified
as sketched in Fig. 2.8. At the front interface, the balance between friction and wetting
forces at the vicinity of the contact line leads to the existence of a non-zero dynamic
contact angle and this flattening of the advancing interface increases the resistance to
the motion. By assuming that θa has the same value in the two sections, the dynamic
capillary pressure drop for the front interface becomes
a
∆Pcap
= −γκ cos θa .
(2.6)
The relation between the contact angle and the velocity of the contact line is known
through Tanner’s law [19, 63] which states that
θa = (6ΓCa)1/3 ,
(2.7)
where Γ is a prefactor (Γ ' 5) that accounts for the singularity occurring at the contact
line [44].
Using Eq. (2.7) and taking the leading-order term in the Taylor expansion of cos θa
in Eq. (2.6), the front pressure drop becomes
!
2/3
(6ΓCa)
a
∆Pcap
' γκ −1 +
.
(2.8)
2
At the rear interface, a thin film is deposited at the channel walls [24], which reduces the
rear meniscus radius and thus also increases the resistance to the motion. The dynamic
capillary pressure drop at the rear interface may then be written as
1
1
r
∆Pcap = γ
+
,
(2.9)
b/2 − eb w/2 − ew
where we denote eb and ew the thicknesses of the deposited film at the rear of the plug
in the thickness and plane sections respectively.
Bretherton’s law expresses the thickness e of the deposited film at the rear of a plug
as a function of the capillary number Ca and the radius R of a circular capillary tube [19]:
e/R = 3.88Ca2/3 .
(2.10)
Assuming Bretherton’s law to be valid in both directions of the rectangular cross section,
i.e. 2eb /b = 2ew /w = 3.88Ca2/3 , the rear capillary jump is obtained from Eq. 2.9 as
r
= γκ 1 + 3.88Ca2/3 .
∆Pcap
(2.11)
Finally, by using Eqs. (2.4), (2.8) and (2.11), one derives
Pdr ' αηL0 U0 + γκβCa2/3 ,
(2.12)
where β = 3.88 + (6Γ)2/3 /2 is a nondimensional coefficient obtained from Bretherton’s
and Tanner’s laws.
43
Equation (2.12) governs the dynamics of a plug of length L0 moving steadily at
speed U0 in a straight rectangular channel for a constant driving pressure. It can be
nondimensionalized by comparing it with the capillary pressure jump γκ, and we obtain
Pdr ' αCa + βCa2/3 ,
(2.13)
with Pdr = Pdr /(γκ) the nondimensional driving pressure and α = αL0 /κ is the nondimensional length.
150µm
AIR
2.2.2
Comparison with experiment
Liquid is
plug
A typical experimental velocity vs. length curve
shown in Fig. 2.9 for a particular
driving pressure (Pdr = 250 Pa). The nonlinear dependence of the velocity on the inverse
length is evident when the experimental data are compared with the dashed line which
represents the prediction for purely viscous flow. This viscous law approximates the travel
velocity well for long plugs, i.e. L0 w, but theLength
difference increases as the length of the
plug decreases and the losses at the interfaces become the dominant mechanism. The
solid line is fitted from Eq. (2.13), using β as a fitting parameter. It yields a value of β
(PFD)
that is consistent with theLiquid
values
obtained in the literature for different geometries [19].
!3
x 10
2.5
2
Ca
1.5
1
0.5
0
0
0.01
1/α
0.02
0.03
Figure 2.9: Capillary number vs. the inverse plug length. The dashed line shows the
Poiseuille law and the continuous line indicates the relation (2.13).
The significance of this result for microfluidics lies in the transport of drops and
bubbles. It has been noted by several groups that the presence of drops increases the
resistance to flow in a microchannel [3, 56, 104], though no clear mechanism has been described for this increased pressure drop. By looking at the space between the drops rather
than the drops themselves, one begins to understand the contribution of the interface deformation on the pressure balance across a train of drops. Although this mechanism does
not account for other effects such as flow in the corners or Marangoni retardation, it
explains the characteristic “bullet” shape that is often published, which is clear evidence
that the change in shape of the front and rear interfaces plays at least a partial role in
retarding droplet traffic.
1
44
2.3
Flow of a plug in a bifurcation
We now consider a plug flowing through a bifurcation with right angles: a T-junction.
We characterize the behavior of this plug for different driving pressures and initial plug
lengths, L0 , measured in the straight channel upstream of the bifurcation. Three different
behaviors may be observed, as described below.
2.3.1
Blocking
The first feature we can extract is the existence of a threshold pressure Pthresh (between
200 & 300 Pa for our experiments) which does not depend on the initial length of the
plug. For Pdr < Pthresh , the plug remains blocked at the entrance of the bifurcation
as shown in Fig. 2.10, which shows the equilibrium positions of a plug for two distinct
pressures, Pdr = 100 Pa & Pdr = 200 Pa. While the rear interface has the same shape
in both cases, the front interface adapts its curvature to balance the applied pressure,
keeping its extremities pinned at the corners of the bifurcation.
C.P. Ody et al. / Journal of Colloid and Interface Science 308 (2007) 231–238
lary number of a liquid plug as a function of its inverse nondingth 1/α for a pressure Pdr = 250 Pa (Pdr = 0.2). The dashed line
o Poiseuille’s law, Ca ∝ 1/α. The full line corresponds to Eq. (11)
Fig. 6. Blocking of the plug at the entrance of the T-junction for two distinct
pressures with Pdr < Pthresh . The front interface is pinned at the corners of the
bifurcation and adapts its curvature to the applied pressure. The shape of the
rear interface is independent of Pdr .
thresh
Figure 2.10: Blocking of the plug at the entrance of the T-junction for two distinct
pressures with Pdr < P
. The front interface is pinned at the corners of the bifurcation
and adapts
its pressure
curvature to the applied pressure. The shape of the rear interface is
= Pdr /(γ κ) is the nondimensional
driving
iments were conducted with channel dimensions wi × wo × b =
L0 /κ is the nondimensional
length.
independent
of Pdr .
260 × 260 × 46 µm. Three different behaviors were observed
ments were conducted in a microchannel to verify
depending on the driving pressure and on the initial length of
ability of Eq. (11) to our rectangular geometry. The
the plug as described below.
=
on of the microchannels used
here was
wi × bresult
This
simple
can nevertheless be of major importance in the case when the
µm. Single plugs of various lengths were introduced
4.1.a Blocking
bifurcation is one link in
network, in which case the distribution of pressures may be
at constant pressure following the protocol of Secthatthat
wetheare
below the threshold value locally, even for a large global driving pressure.
all driving pressures, such
we observed
measured
The first feature extracted from the observations of a plug
d speeds of the plugs remained constant within 5%
getting
to the T-junction is the existence of a threshold prest the channel and average length L0 and speed U0
sure Pthresh (between 200 and 300 Pa for our experiments)
mined.
which does not depend on the initial length of the plug. For
le of experimental results is plotted in Fig. 5. Data
Pdr < Pthresh , the plug remains blocked at the entrance of the
ding to long plugs asymptote to Poiseuille’s law
bifurcation
as presented
in Fig.
6, which shows
the equilibWhen
the
pressure
exceeds
Pthresh
, the plug
continues
its propagation
through the bifurne, Pdr = αCa from Eq. (2)). For shorter plugs, the
rium positions of plugs for two distinct pressures (Pdr = 100,
−1
relationship
to an adit a nonlinear Ca vs (α)
cation.
If thedueplug
is short,
its rear interface catches up with its front interface before the
200 Pa). While the rear interface has the same shape in both
ssipation occurring at the interfaces. The theoretical
latter reaches the opposite
The
plug
then
ruptures,
opening
cases,wall.
the front
interface
adapts
its curvature
to balance
the ap- the outlet branches of
line) obtained from Eq. (11) is plotted with β = 16
plied
pressure sequence
keeping its extremities
pinned
at the corners
of
the
T-junction
to
air.
A
typical
of
plug
rupture
is
shown
in Fig. 2.11. Evenood agreement with the experimental data. The corthe bifurcation.
g value of Γ = 20 is higher
than
experimental
values
tually, the liquid that remains on the channel walls drains slowly through the action of
n millimetric circular tubes [1]. However, the coeffi4.2. Rupture
capillary
forces
and
Tanner’s law
(Eq. (5))
is inair drag.
)1/3 = 4.9 appearing in
ement with Hoffman’s data [11] for which the previWhen the pressure exceeds the threshold pressure Pthresh , the
ient is of order 4–5 for a dry tube. A more precise es45through the bifurcation. If the
plug continues its propagation
he Tanner’s constant would require a specific study by
plug
is
short,
its
rear
interface
catches up with its front interface
ack of both the microscopic and macroscopic physics
before the latter reaches the wall opposite to the entrance chann our configuration, as described in [12].
nel. The plug then ruptures, opening the outlet branches of the
although Bretherton’s and Tanner’s laws are not
T-junction to air. A typical sequence of plug rupture is shown in
plicable to the rectangular geometry, our results show
Fig. 7. Eventually, the liquid that remains on the channel walls
lity of using the previous laws to model the dynamics
drains slowly through the action of capillary forces and air drag.
our channels.
2.3.2
Rupture
235
235
Fig. 7. Sequence of plug rupture. The plug ruptures in the T-junction, leaving liquid on one side of the outlet channel, where air can flow freely from the pressure
Figure 2.11: Sequence of plug rupture. The plug ruptures in the T-junction, leaving
source.
Fig. 7. Sequence of plug
rupture.
The plug
the T-junction,
leaving
liquid
one side
the outlet
channel,
air can source.
flow freely from the pressure
liquid
on one
sideruptures
of thein outlet
channel,
and
aironcan
flowoffreely
from
thewhere
pressure
source.
Fig. 8. Sequence of plug splitting at the T-junction. Splitting leads to symmetric formation of two daughter plugs in the outlet branches of the T-junction.
Figure
2.12: Sequence
of plug
splitting
at the T-junction. Splitting leads to symmetric
dynamics becomes
dominated
by the wetting
forces
acting beformation
of
two
daughter
plugs
in
the
outlet
branches of the T-junction.
tween the becomes
two phases.
These forces
to draw
the liquid
dynamics
dominated
by theact
wetting
forces
actingvery
berapidly
into
theforces
wall,act
as to
observed
the
ween the
twocontact
phases.with
These
draw thebetween
liquid very
images (b)
(c) ofwith
Fig.the
8. In
partas(c),
we see between
that the currapidly
intoand
contact
wall,
observed
the
2.3.3
Splitting
into implying
two daughter
plugs
vature
is
not
constant
along
the
liquid
surface,
mages (b) and (c) of Fig. 8. In part (c), we see that the local
curpressure
in the
pull
it into
the
vature
is gradients
not constant
along
thewhich
liquid
surface,
implying
local
When
thefluid
plugs
are sufficiently
longdaughter
that
the front interface reaches the opposite wall, two
branches.gradients
Thesedaughter
wetting
are
dominant
compared
to the through a splitting of the initial one, as shown
pressure
in the forces
fluid
pull length
it into
the
plugswhich
of equal
are daughter
formed
driving pressure,
as seenforces
by theare
rapidity
of the
advancetoonce
branches.
These in
wetting
dominant
compared
the the wall, the transport dynamics becomes
Fig. 2.12. As
soon
as the
liquid reaches
the
plug
has
touched
the
wall.
driving pressure,dominated
as seen by by
thethe
rapidity
of the
advance
once
wetting
forces
acting
between the liquid and solid. These forces act to
before rupture
or splitting, the evolution of the plug
he Note
plug that
has touched
the
wall.
draw the liquid very rapidly into contact with the wall, as observed between the images
in Note
the bifurcation
isrupture
similar
both cases:
the frontofinterface
that before
or in
splitting,
the evolution
the
plug
(b)constraints
and (c)
of
Fig.the2.12.
In part
(c),
we see that the curvature is not constant along
advances
without
while
extremities
ofinterface
the rear
n the bifurcation
is
similar
in
both
cases:
the
front
the
liquid
surface,
implying
the
existence
local pressure gradients which pull the liquid
interface remain
pinned
by thewhile
corners
of
the bifurcation
until
advances
withoutinto
constraints
the
extremities
of
the
rear
daughter
branches.
These wetting forces are dominant compared to the driving
process
rupturepinned
or the
splitting
iscorners
achieved
(Fig.
7c).
nterfaceofremain
by
the
of
the
bifurcation
pressure, as seen by the rapidity of theuntil
advance once the plug has touched the wall.
process of rupture or splitting is achieved (Fig. 7c).
Note
that
before
rupture
or
splitting,
the evolution of the plug in the bifurcation is
4.4. Phase diagram
similar in both cases: the front interface advances without
constraints
while
the extremFig. 9. Phase
diagram for
plug behavior
at the T-junction.
4.4. Phase diagram
ities
the rear
by the cornersFig.
of 9.the
bifurcation
until
process
We summarize
theofthree
casesinterface
discussedremain
above pinned
(blocking,
Phase
diagram for plug
behavior
at theof
T-junction.
rupture
or
splitting
is
achieved
(Fig.2.11-c).
2D
pressure
balance
to
derive
the
value
of
the
blocking presrupture
and
splitting)
in
the
experimental
phase
diagram
shown
We summarize the three cases discussed above (blocking,
sure
after
which
we
develop
a
model
to
predict
the transition
in
Fig.
9.
As
stated
above,
we
observed
the
blocking
(!)
of
the
The
three
behaviors
can
be
summarized
in
the
phase
diagram
of
Fig.
2.13
as
a
2D pressure balance to derive the value of the blocking
presrupture and splitting) in the experimental phase diagram shown
between
rupture
and
splitting.
plug
below
a
threshold
pressure,
P
,
whose
value
is
seen
to
function
of
plug
length
and
driving
pressure.
Below,
we
will
develop
a
physical
picture
thresh
sure after which we develop a model to predict the transition
n Fig. 9. As stated above, we observed the blocking (!) of the
be independent
on
the length
of the
plug.
Abovevalue
Pthresh
expercorresponding
to
the
between
regions.
between
rupture and splitting.
plug
below a threshold
pressure,
Pthresh
, transitions
whose
is ,seen
tothe three
5.1.
Threshold
pressure
imental
data
show
that
the
transition
between
rupture
(!)
and
be independent on the length of the plug. Above Pthresh , expersplitting
(!) show
depends
thetransition
applied pressure
on the
length
mental data
thatonthe
betweenand
rupture
(!)
and 46 5.1. Threshold pressure
Since the plug is at rest in the blocking case, the viscous term
of
the
plug:
as
the
pressure
increases,
the
length
of
liquid
resplitting (!) depends on the applied pressure and on the length
in Eq.
(1)the
is plug
zero is
and
the new
balance
may
be written
quired
to create
twopressure
daughterincreases,
plugs becomes
smaller.
Since
at rest
in thepressure
blocking
case, the
viscous
term
of
the plug:
as the
the length
of liquid reas
in Eq. (1) is zero and the new pressure balance may be written
quired to create two daughter plugs becomes smaller.
5. Theoretical model
r
a
as
+ !Pcap
.
Pdr = !Pcap
(12)
237
hat
2#
!
"
$ %
$ %&
π
f θa − g θr − 1 −
.
4
(19)
tact angle θ a , a plug of initial length L0 < Lcrit
he bifurcation while a plug such that L0 > Lcrit
o daughter plugs.
a function of the ratio wo /wi , the value of Lcrit
and wo /wi . Therefore, Eq. (19) will provide diff Lcrit as θ a varies, the other quantities being
given geometry. The problem is now to underanisms involved in the determination of θ a .
c condition
e propagation in the bifurcation is unsteady, the
Fig. 12. Experimental Pdr –L0 /wi diagram and theoretical transitions. (1) Rupll governed by a visco-capillary regime. Thereture-splitting
transition corresponding
to Eq. (27),
withsplitting
θ a varying from
Phase
showing
the breaking
◦,π/6
and blocking 4 regimes.
n time, we may use Figure
Eq. (1) on2.13:
a specific
fluid diagram
to π/3 and (6Γ )1/3 = 4.9. (2) Threshold pressure (Eq. (14)).
Thethelines
correspond
to the theoretically predicted values. The solid lines correspond to
simplicity, we choose
streamline
linking
the theoretical model developed
in thethetext.
Finally, combining
terms above leads to the pressure bal-
nt C, the speed of the rear interface is nil since it
corners of the junction. Thus, the capillary drop
interface is approximated as
"
2
+
,
(20)
b
The two parameters
2.3.4
ance across the plug
'
(
%
yE − wi /2 $ a %3
1
1
2$
a
Pdr " γ 2
θ
+
+
+
,
1
−
cos
θ
Ra Rr b
Γ b2
Modeling the transitions
(26)
that
determine the transitions between the three behaviors observed
which accounts for the dynamic relation between θ a and Pdr .
e in-plane radius of above
the receding
interface
(see
are plug length and
the transition
Usingforcing
a referencepressure.
pressure γ /wFirst,
the different between the blocking
i and expressing
expressed as
a , w /w , w /b and w /b, Eq. (26)
quantities
as
functions
of
θ
i
o
o
i
and flowing can be understood by writing a pressure balance on the two sides of the plug
reads in a nondimensional form
.
as it passes the (21)
bifurcation,
into
the change in curvature at each
r
' while
! "account
! taking
"
(
wo w i
wi 2 (θ a )3
θ a −1
interface.
assume
that
front interface
remains pinned to the corners, we can
Pdr "
2 the tan
−
terface, we assume that
the speed ofIf
thewe
contact
b b
2
b
Γ
a is the
and that
θ
stablishes the contactwrite
angle θ athe
maximum capillary
that "(
must be overcome which corresponds to a
' pressure
!
ckness and in the plane. Therefore, the capillary
wi
wi
radius of curvature equal to
+ w/2,
1 + cos where
2 arctan w is the width of the entrance channel. In this
ancing interface reads
2wo
wo
case, the threshold pressure 'can be written
as
(
a"
−
2 cos θ
b
,
(22)
in-plane radius of the advancing interface, calometry as
+
%
2wi
wi $
+
1 − cos θ a .
b
wo
Pthresh
4γ
,
=
w
(27)
(2.14)
5.2.3. Summary of the model
The geometrical condition provides a relation between Lcrit
which is shown by
line (2) on Fig. 2.13.
.
(23) the solid
a
and θ a (Eq. (19)) while θ a may be related to Pdr using the
In order to understand
the transition
between
splitting
pressure balance
of Eq. (26). For
a given value
of θ a , weand
are rupture, we consider
rm in Eq. (1) is approximated as a dissipation
now
able
to
estimate
the
value
of
P
corresponding
to
L
.
By
dr
crit
the
interface exactly at
leadinginto which thea rear interface catches up with the front
Shaw geometry (since
b/wcritical
o " 0.18),case
varying θ from π/6 to π/3, we obtained the results plotted in
the moment when the their
front
interface
reaches
the
opposite
wall,
as
shown
in Fig. 2.14(b).
dimensionless form in Fig. 12, using Γ = 20 which is the
Vm ,
(24)
In this case, we may reduce
the problem
a geometric
which all the parameters
value obtained
from straightto
channel
experiments.one
The in
quantitative
and
qualitative
behaviors
of
the
theoretical
transition
are
e length of the streamline
C–E
and
V
the
mean
m
are fixed by the channel design, except the contact angle at the front
interface (point E ).
in good agreement with the experimental data.
uid particles between points C and E. As stated
This
angle
contains
the
physics
of
the
problem,
since
it
varies
with
the velocity of the
ed at point C is nil and the speed at point E is
advancing
interface,
through
a visco-capillary balance.
ontact angle by Tanner’s
law (Eq. (5))
as
6. Conclusion
By assuming that the angle variations are given by Tanner’s law as for the case
Our experiments on plug propagation in straight microchan(25)
of a straight channel, nels
oneofmay
again
obtain
relation
between
rectangular
cross
section the
showed
good agreement
with the pressure drop and
previous
studies
capillaries
with circular
sections
[1], in spite
e quantities Vm and the
Lm toliquid
first order
as Vm = This
velocity.
relies
on inan
evaluation
of the
viscous
contribution, which must
of the large aspect ratio of our channels. In particular, the two
nd Lm = yE − wi /2, where yE = wo / tan(θ a /2)
be
estimated
since
the
details
of
the
flow
field
are
unknown.
However, a mean value
principal radii of curvature could be used to account for the
rom point E to the centerline of the inlet channel
corresponding to grosscapillary
averaging
ofthethe
lengthandand
velocity
of and
the fluid can be used to
effects at
advancing
receding
interfaces
yield a relation that uniquely relates the driving pressure and the contact angle. The
critical volume to reach the configuration of Fig. 2.14(b) can therefore be computed as a
function of driving pressure. This prediction corresponds to the line (1) on Fig. 2.13 and
agrees well with the experimental observations.
γ (θ a )3
.
6ηΓ
47
of the rear and front interfaces at the threshold pressure
ributions of the capillary pressure jumps which
(a)
(b)
(c)
aplace’s law to the corresponding principal curterfaces. Since the channel thickness does not
Fig. 11. Shape of the plugs before (a) rupture and (c) splitting. (b) Schematic
cation, we assume that
the in-thickness
com- of of
a plug
of volume
Vcrit situated
on the theoreticaland
boundary
between rupture
Figure
2.14: Shape
the
plugs
before
(a)-rupture
(c)-splitting.
-(b) Schematic of a
and splitting where front and rear interfaces of a plug meet at the wall.
pressure drops at the rear and front interfaces
plug
of thickness
volumebeing
Vcrit situated on the theoretical boundary between rupture and splitting
r, the radii of curvature
in the
and torear
of acondition
plug meet at the wall.
value. Therefore, thewhere
problemfront
is reduced
a interfaces
5.2.1. Geometric
nce of in-plane components of pressure drops,
The diagram of Fig. 9 exhibits a L0 –Pdr dependent boundary
between the rupture or splitting cases. We consider the shape of
the plugs just before rupture or splitting, as shown in Figs. 11a
"
1
and 11c. In Fig. 11a, the rear interface catches up with the front
(13)
,
Ra
the latter reaches the wall, leading to rupture
This work was interface
done inbefore
collaboration
with Jean-Baptiste Masson and Xin
a are the radii of curvature of the rear and front
of the plug. Conversely, in Fig. 11c, the front interface is about
Wang
[15].
to touch the wall before the rear interface catches up with it,
ne A, respectively.
resulting in the splitting of the plug and the formation of two
ental observations shown in Fig. 6 provide the
daughter
plugs. In the
limiting case,
of length behavior
Lcrit sees
Most
recently,
we
addressed
questions
ofa plug
collective
that comes about
erive the value of the threshold
pressure.
For
its
front
and
rear
interfaces
meeting
at
the
wall.
Approximating
e receding interface through
is seen to keep
a constantinteractions at the exit of parallel channels. In particular, we looked at
capillary
the shape of both interfaces with circular sections leads to the
of curvature R r = wi /2 while the advancing
the flow of an oil interface
exiting
a thin
a issudden
sketch of
Fig. 11b.from
Here, the
criticalchannel
volume of into
the plug
noted expansion for three
its in-plane curvature to the applied pressure
a , θ r , w and the thickness b
and
is
completely
defined
by
θ
V
a varies such that |R
a
crit
o
microchannel
geometries: (a) Flow through a single channel,r (b) through two parallel
| ! wi /2. The situation
of the microchannel. At point C, the rear contact angle θ is de+wi /2 is illustrated in
Fig. 10 andor
leads
the
channels,
(c)to through
seven parallel channels.
fined geometrically as θ r = 2 arctan(wi /2wo ). In contrast, the
re, which is the highest pressure that can be
a at the
This work was motivated
to different
understand
the flow through a simple
pointdesire
E may take
values dependcontact angle θby
the plug, reading finally
ing onathe
velocity fluid
of the advancing
interface.adjacent
As a result, pores
the
porous medium, in which
viscous
exits
through
and spreads while
critical volume decreases as θ a increases for a given T-junction,
being submitted(14)
to a visco-capillary
balance. The collective effects of flow through parallel
down to a zero volume when θ a = π − θ r .
pores can thus be explored
and compared
withbetween
singlethepore
models.
The volume
of liquid contained
two interfaces
in
mental conditions (wi = 260 µm and γ =
the critical
configuration corresponding
tothrough
Fig. 11b is a
obtained
The
base
state
which
corresponds
to
the
flow
single
channel is shown in
obtain Pthresh ! 230 Pa, in agreement with the
as
fig.Pa).
2.15(inset), which shows a# superposition
of the interface shape at successive times.
(200 Pa < Pthresh < 300
$ a%
$ r %&
2
f
θ
−
g
θ
,
=
bw
V
crit be oobserved from these experiments: (15)
Two important facts may
First, the visco-capillary
splitting
balance is verified throughout
the experiment, which is shown by measuring the flowrate
where
$ a % $ a the channel
%$ dimensions
%−2
a function
time.
were chosen
pose to explain the as
critical
plug length,ofLcrit
,
f Indeed,
θ = θ − cos θ a sin θ a 1 −
(16) such that the main
cos θ a
ure or splitting whenpressure
the pressuredrop
exceeds
the
occurs
in
the
long
and
thin
“impedance”
channel,
which yields a constant
and
re. Our aim here is not
to
model
the
complete
flowrate at late times. However,
the capillary%$ effects %are
significant at intermediate times
$ % $
−2
plug but to derive a global model which clarg θ r = π − θ r + cos θ r sin θ r 1 + cos θ r .
(17)
t =pressure
2 s in PFig.
2.15), where a dip in the flowrate appears due
to the large curvature
ence of Lcrit on the(at
driving
dr by
unt the fundamentalwhich
mechanisms
involved
resists
flow.in This volume corresponds to a plug of length Lcrit in the entrance
channel such that
n. We first consider a plug ofSecond,
length Lcritthe
theo-interface
displays
a parabolic,
rather than circular shape. This “lubri&
#
on the transition curve. A geometric condition
= b Lcrit wiin
+ (1
− π/4)wi2 , studies [93], although
Vcrit
(18)
cation
limit”
has
been
predicted
imbibition
the physical origin
from the shape of such a plug in the T-junction.
2
and
applicability
of
the
in our
particular
geometry
are not yet fully
ic argument is used to
mapthe
this geometric
conwith
the lubrication
correction term limit
b(1 − π/4)w
into account
the
i taking
xperimental L0 –Pdr understood.
diagram.
volume
contained
in
the
circular
menisci.
Therefore,
the
critical
The parabolic shape may be verified by measuring the maximum width
2.4
Parallel channels
transverse to the flow direction, dx, and the position of the apex in the flow direction zm .
The dependence of zm with dx is linear for the case of a parabolic interface, as verified
by the circles in Fig. 2.16.
48
4
x 10
7
Flow direction
Flowrate ( µm3 /s)
3
2
1
0
0
2
4
time (s)
6
8
Figure 2.15: Evolution of the flowrate as the fluid exits through a single channel. The
driving pressure is Pdr = 785 Pa. The image shows the location of the oil interface at a
succession of times. For clarity, the time step used in the display is dilated as the liquid
exits the channel, as seen by the apparent jumps in the interface position. The time step
is dt = 0.1 s for the first period, dt = 0.2 s for the second, and dt = 0.4 s for the third.
Next, this base state is compared with the flow through two parallel channels separated by a distance of d = 200, 300, or 500 µm. Through similar measurements of the
interface shape and position (Fig. 2.16), we uncover the existence of three regimes:
1. At early times, the two separate drops act as independently advancing droplets.
This is confirmed in Fig. 2.16(b) where the zm (dx − d) shows exact collapse of the
data for different channel separations.
2. When the two drops begin to interact, a very rapid merging event occurs, which we
do not resolve with our experiments, followed by a slower relaxation to the parabolic
shape. This relaxation occurs through the redistribution of the flow into the central
region between the two exiting droplets. It is also associated with a temporary
increase in the total flowrate. In Fig. 2.16(a), the merging event corresponds to the
kink in the curves, meaning that the transverse flow is strongly reduced compared
to the flow in the axial direction.
3. Finally, a parabolic evolution similar to the case of the single channel is recovered at
late times. The time taken to reach the parabola can be predicted by visco-capillary
analysis and scales as d3 , which is verified in the experiments.
The interface shape before, during, and after the merging can be seen in Fig. 2.16(c),
which gives a visual verification of the global quantities discussed above.
The transient state plays a major role in the case of n parallel channels, as shown
in Fig. 2.17, which shows a superposition of images corresponding to the flow through
seven parallel channels. While the early and late times reflect again the results obtained
in the two-channel case, we observe that the mergings happen mainly on a two-by-two
49
600
400
zm (µm)
(b)
zm / tan(!) (µm)
500
(a)
300
200
100
400
0
0
200
400
600
dx ! d (µm)
300
200
(c)
100
0
0
200
400
600
800
1000
1200
1400
1600
1800
dx (µm)
Figure 2.16: (a) Evolution of the modified maximum position (zm / tan(θ)) with respect
to the lateral width of the interface (dx) for a single channel (◦) and for two channels
separated by 200 µm (∗), 300 µm (), and 500 µm (4). The straight line represents
zm = dx/4. (b) Same data with the inter-channel distance (d) subtracted. (c) The
interface before, during, and after the transition.
basis among exiting drops. This is due to the interactions between adjacent droplets
which, when they merge, redirect the fluid to the space between them, thus temporarily
preventing further mergings from occurring. For this reason, the transition between
n independent drops and a single parabolic shape occurs through a cascade of binary
mergings and relaxations. This implies that the channel doublet is the fundamental
building block which may be generalized to any number of total channels.
Figure 2.17: Advance of an interface through seven independent channels. The time
separating each line is 0.1 s.
50
c 2005 Cambridge University Press
J. Fluid Mech. (2006), vol. 546, pp. 285–294. doi:10.1017/S0022112005007287 Printed in the United Kingdom
285
The propagation of low-viscosity fingers
into fluid-filled branching networks
By C H A R L E S N. B A R O U D1 , S E D I N A T S I K A T A1
A N D M A T T H I A S H E I L2
1
Laboratoire d’Hydrodynamique (LadHyX), Ecole Polytechnique, 91128-F Palaiseau Cedex, France
2
School of Mathematics, University of Manchester, Manchester, M13 9PL UK
(Received 12 January 2005 and in revised form 20 June 2005)
We consider the motion of a finger of low-viscosity fluid as it propagates into a
branching network of fluid-filled microchannels – a scenario that arises in many applications, such as microfluidics, biofluid mechanics (e.g. pulmonary airway reopening)
and the flow in porous media. We perform experiments to investigate the behaviour of
the finger as it reaches a single bifurcation and determine under what conditions the
finger branches symmetrically. We find that if the daughter tubes have open ends, the
finger branches asymmetrically and will therefore tend to reopen a single path through
the branching network. Conversely, if the daughter tubes terminate in elastic chambers,
which provide a lumped representation of the airway wall elasticity in the airway
reopening problem, the branching is found to be symmetric for sufficiently small
propagation speeds. A mathematical model is developed to explain the experimentally
observed behaviour.
1. Introduction
Flows in which a finger of low-viscosity fluid propagates into a tube and displaces
another fluid of larger viscosity are of relevance in many applications, such as oil
extraction (Homsy 1987), microfluidic devices for droplet transport (Link et al. 2004)
and pulmonary biomechanics (Gaver, Samsel & Solway 1990).
For instance, many respiratory diseases, such as respiratory distress syndrome, may
cause the occlusion of the pulmonary airways with viscous fluid. Occluded airways
are believed to be reopened by a propagating air finger, in a process that involves a
complicated interaction between a viscous free-surface flow and the deformation of
the elastic airway wall. The mechanics of airway reopening in an individual airway
have been investigated by many authors (e.g. Gaver et al. 1990; Hazel & Heil 2003),
but these studies ignore the fact that the pulmonary airways branch frequently.
This raises the question of whether the propagating air finger will reopen the entire
pulmonary tree or simply follow a single path, keeping most of the lung occluded.
(Cassidy, Gavriely & Grotberg (2001) performed experiments to determine how short
liquid plugs propagate through rigid bifurcations.) The question is also of relevance in
‘gas embolotherapy’, a novel technique aimed at starving tumour cells of their blood
supply (see Calderon & Bull 2004).
Similar questions arise in microfluidic technology. Many microfluidic devices are
designed to transport samples of fluid through complicated networks of channels in
order to perform chemical or bio-chemical tests (Song, Tice & Ismagilov 2003).
The use of drops as vehicles for the transport offers many advantages, such as
286
C. N. Baroud, S. Tsikata and M. Heil
(a)
L1(t)
V1(t)
From
syringe pump
)
U 1(t
Q
pdr(t)
Microscope
(b)
Membrane
Channel
U (t
2 )
PDMS
body
L2(t)
100 µm
~1cm
V2(t)
Glass slide
Figure 1. (a) Model problem: low-viscosity fluid is injected into a branching tube. The
propagating fingers displace some of the more viscous fluid that initially filled the tube; the
remaining viscous fluid is deposited on the tube walls. The two daughter tubes are terminated
by large elastic chambers. (b) Sketch of the experimental set-up (not to scale). The total volume
of fluid contained in the channel is approximately 0.5 µl while the volume in the end chambers
is approximately 50 µl.
the control of dispersion and wall contact, as well as the possibility of activating
chemical reactions at controlled locations, e.g. through droplet merging. Other uses
of bubbles and drops in microfluidics include the production of controlled microemulsions through the breakup of long bubbles as they advance through a branching
network of microchannels (Link et al. 2004). It is therefore important to understand
the dynamics of the drop transport in order to predict the path and the breakup of
individual drops.
Here, we study these questions at the level of a single bifurcation and consider
the model problem shown in figure 1(a). A finger of low-viscosity fluid propagates
along a parent tube which branches into two daughter tubes which have identical
cross-sections. The finger is driven by the injection of fluid at a constant flow rate, Q,
and displaces a viscous fluid of viscosity µ which is much larger than the viscosity of
the finger. We denote the surface tension between the two fluids by σ . The daughter
tubes terminate in two large elastic chambers which provide a lumped representation
of the wall elasticity in the pulmonary airways. We wish to establish the conditions
under which the finger continues to propagate symmetrically along the two daughter
tubes once it has passed the bifurcation.
2. Experimental set-up
We performed experiments using the set-up sketched in figure 1(b). A Y-shaped
channel of rectangular cross-section (W × H = 200µm × 100 µm) was fabricated using
standard microfluidic soft lithography methods (Quake & Scherer 2000). A mould
was made by etching a 100 µm layer of photoresist on the surface of a silicon wafer.
The mould was then covered with a thick layer of liquid poly(dimethylsiloxane)
(PDMS) which was allowed to solidify before we created cylindrical chambers at
the ends of the three branches: a thin hole at the end of the parent branch for
the fluid injection, and two larger end chambers (holes with 1.6 mm radii) at the
ends of the two daughter channels. The large end chambers were sealed with thin
elastic membranes of thickness h = 50 µm, also made of PDMS (Young’s modulus
E 9 × 105 N m−2 and Poisson’s ratio ν = 0.5). Finally, the PDMS block was bonded
onto a glass microscope slide.
Finger propagation in branching networks
an
Br
287
ch1
Q
Br
anc
h2
400 µm
Figure 2. Snapshot of the branching finger. The positions of the two tips are recorded over
a sequence of successive images, allowing the velocity of the fingers to be determined.
The experiments were conducted by first filling the channels with silicone oil
(viscosity µ = 100 cP) until the end membranes were taut. PDMS is permeable to
gas, therefore the air that initially filled the channels could escape through the thin
membranes when the silicone oil was injected. After the membranes were slightly prestretched, a finger of perfluorodecalin (PFD; viscosity µPFD = 2.9 cP) was injected.
This was done using a syringe pump with a 100 µl glass syringe, at flow rates in
the range 1–60 nl s−1 , resulting in finger velocities of 50–3000 µm s−1 in the main
branch. The choice of fluids was guided by the requirements that the two fluids be
incompressible and immiscible, and that they wet the PDMS and the glass surfaces.
The surface tension between the two fluids is approximately σ = 50 mN m−1 .
Image sequences of the advancing fingers were taken through a stereo-microscope
at × 7.11 magnification, using a digital camera at 2000 × 2000 pixel resolution. A
typical image is shown in figure 2. We tracked the position of the advancing fingers
in the two daughter tubes over a sequence of images to determine the speed of the
two finger tips.
3. Experimental results
Figure 3 shows the evolution of the finger velocities in the two daughter tubes
for fingers that are driven by the injection of fluid at two different flow rates. At
low flow rates the fingers propagate at approximately the same speed, resulting in
an approximately symmetric branching pattern. Conversely, at higher flow rates, any
small initial difference in the finger speeds increases rapidly as the fingers propagate
into the daughter tubes, resulting in strongly asymmetric branching. We observe
that in both cases the sum of the two finger speeds remains constant within experimental error. Experiments conducted in open-ended channels, i.e. without the elastic
membranes, always displayed strongly asymmetric propagation, regardless of the
driving flow rate. Finally, we note that the initial filling of the viscous oil into an
air-filled channel always occurred in a symmetric manner, both with and without the
elastic membranes. This is consistent with the experiments of Cassidy et al. (2001) who
found that finite-length plugs of oil branched symmetrically when flowing through a
symmetric air-filled bifurcation.
288
(a)
(b)
500
Velocities (µm s–1)
2000
400
1500
300
1000
200
500
100
0
20
40
Time (s)
60
0
5
10
15
Time (s)
Figure 3. Time series of the velocity of the advancing fingers in channels 1 (䊊) and 2 (×) as a
function of time, for two different forcing flow rates. The sum of the two velocities (*) remains
approximately constant.
Velocity in parent branch, Up
61.78 µm s–1
107.3
125.0
‘Symmetric’
252.6
478.8
606.7
767.8
‘Asymmetric’
941.6
1881
300
U1 – U2 (µm
s–1)
450
‘Asymmetric’
150
‘Symmetric’
0
0
0.2
0.4
0.6
Normalized time, tUp/2Lc
0.8
1.0
Figure 4. Evolution of the velocity difference for the fingers in the two daughter branches.
At low flow rates, the velocity differences remain constant. When the velocity in the parent
branch exceeds a threshold, the difference in the velocities increases rapidly. We refer to the
two regimes as ‘symmetric’ and ‘asymmetric’, respectively.
Results for channels with elastic chambers are presented in figure 4 where the
symmetry of the branching is characterized by plotting the difference between the
finger velocities as a function of the ‘normalized distance’ tUp /2Lc , where Lc = 1.7 cm
is the length of the daugther channels. This ordinate facilitates a direct comparison
between the different experiments whose duration varies significantly.
At small flow rates, the difference in finger speeds remains approximately constant.
When the velocity in the parent channel becomes larger than a threshold, the velocity
difference increases rapidly as the fingers propagate along the daughter branches – the
289
experiment ends when the ‘faster’ finger reaches the downstream end of its daughter
tube. For a given microchannel, the faster finger is always associated with the same
branch, suggesting that small imperfections in the fabrication process produce the
initial perturbation. These initial perturbations are strongly amplified at high flow
rates but remain almost constant if the flow rate is below a threshold. For simplicity,
we will refer to the two regimes as ‘asymmetric’ and ‘symmetric’, respectively.
The signature of the asymmetric regime is also observed when the low-viscosity
finger is not continuous, but made up of discrete drops. These drops may be produced
by a splitting of the PFD finger at the entrance of the microchannel, for example if the
connecting tube is lined with silicone oil. As the train of drops reaches the bifurcation,
each is split into two ‘daughter droplets’ which travel down the daughter channels.
The length ratio of the two daughter drops depends on the relative velocity in the two
branches. In the symmetric case, the size ratio remains constant as successive drops
arrive at the bifurcation, indicating a constant velocity difference. In the asymmetric
case, the velocity difference increases and so does the length ratio of the divided
daughter drops. This ratio may eventually diverge as the fast finger approaches the
end of the channel and the flow in the other branch slows to a halt.
4. Analysis
We will now develop a simple mathematical model to explain the experimental
observations. Since the finger is driven by the injection of fluid at a constant flow rate
Q, the pressure, pdr (t), required to drive the finger at this rate varies with time. We
denote the lengths of the fluid-filled parts of the daughter tubes ahead of the finger tips
by Li (t) (i = 1, 2) and assume that the daughter tubes have the same cross-sectional
area, A.
As the fingers propagate, they displace most of the viscous fluid that initially filled
the tube; far behind the finger tips, a thin stationary film of the more viscous fluid is
deposited on the tube walls (see Taylor 1961). We denote the flow rate in daughter
tube i by Qi = Ui Ai , where Ai is the cross-sectional area occupied by the finger
which propagates with speed Ui . In the absence of inertial and gravitational effects,
a reasonable approximation for microfluidic devices, we have Ai = A α(Ui ), where
the function α(Ui ) has been determined for many tube shapes (e.g. Bretherton (1961)
and Reinelt & Saffman (1985) for circular tubes; Wong, Radke & Morris (1995)
for polygonal tubes; and Hazel & Heil (2002) for tubes of elliptical and rectangular
cross-section – we will refer to these references collectively as ‘R’). Conservation of
mass requires that Q1 + Q2 = Q.
In each of the daughter tubes, the pressure drop across the occluded section has
three components: (i) the Poiseuille pressure drop ahead of the finger tip, ppois =
µRLi (t)Qi (t), where the flow resistance R depends on the tube’s cross-section. For
instance, for a circular tube of radius a, we have R = 8/(πa 4 ); values for other
cross-sections can be found in the literature (e.g. White 1991); (ii) the pressure drop
across the curved tip of the finger, ptip = C(Ui ), where the function C(Ui ) for many
tube shapes is available from the references R (we note that ptip includes capillary
and viscous contributions); (iii) the pressure pelast in the chambers at the end of the
daughter tubes. Provided the membranes that close the (otherwise rigid) chambers
only on the chambers’ instantaneous volume and we assume
are elastic, pelast depends
t
that pelast i (t) = k 0 Qi (τ ) dτ + p0i , where k is a constant and p0i is the pressure in
chamber i at time t = 0.
290
Assuming that the viscous pressure drop in the fingers can be neglected, the fingers
in both daughter tubes are subject to the same driving pressure pdr (t), and we have
t
Qi (τ ) dτ + p0i for i = 1, 2,
(4.1)
pdr (t) = C(Ui ) + µRLi (t)Qi (t) + k
0
where Qi (t) = AUi (t)α(Ui (t)). Differentiating the two equations in (4.1) with respect
to t yields
dpdr (t)
dUi (t) dC(U ) dLi (t)
=
α(Ui (t))Ui (t)
+
kAα(U
(t))U
(t)
+
µAR
i
i
dt
dt
dU U =Ui (t)
dt
dUi (t) dα(U ) dUi (t)
+ Li (t)
Ui (t) + Li (t) α(Ui (t))
for i = 1, 2.
dt
dU dt
U =Ui (t)
Together with the two equations Ui = −dLi /dt (for i= 1, 2), this provides a system
of four ordinary differential equations, augmented by the algebraic constraint
U1 (t)α(U1 (t)) + U2 (t)α(U2 (t)) = Q/A. Thus, we have five equations for the five
unknowns pdr , U1 , U2 , L1 and L2 . If p01 = p02 , these equations admit the symmetric
solution Li (t) = l0 − Ut, where the finger velocity Ui = U in both daughter tubes is
given implicitly by
1Q
Uα(U) =
.
(4.2)
2A
i (t),
We determine the stability of this solution by writing the velocities as Ui (t) = U+ U
where 1, with similar expansion for all other quantities. A straightforward linear
2 (t) = 0 and d
1 (t) + L
pdr /dt = 0 for the
stability analysis then yields the relations L
perturbed quantities. The velocity perturbations are governed by
Ui (t)
t
du
UF(U)
=
dτ,
(4.3)
i (t=0) u
U
0 G(U, τ )
where
and
dα(U ) 1 dC(U ) U +
G(U, t) = (l0 − Ut) α(U) +
dU U
µRA dU U
(4.4)
k
dα(U ) 1−
+U
.
(4.5)
dU U
µUR
i decay, indicIf the right-hand side of (4.3) is negative, the perturbation velocities U
ating that the symmetrically branching solution is stable. To analyse equation (4.3),
we first consider the simplifications α ≡ 1 (ignoring the presence of the fluid film that
the advancing fingers deposit on the channel walls) and C ≡ 0 (ignoring the pressure
jump over the air–liquid interface). In this case, (4.3) simplifies to
t
Ui (t)
du
2U − k/(µR)
=
dτ.
(4.6)
l0 − Uτ
i (t=0) u
U
0
F(U) = α(U) 2 −
k
µUR
The denominator of the integrand on the right-hand side represents the instantaneous
length of the fluid-filled part of the daughter tubes and is therefore always positive.
Perturbations to the symmetrically branching finger will therefore grow if U exceeds
the critical value Uc = k/(2µR). Hence at small velocities, when the flow resistance is
dominated by the vessel stiffness, the finger will branch symmetrically. Conversely, at
291
1.0
α
α + dα/dCa Ca
α, α + dα/dCa Ca
0.8
0.6
0.4
10–3
10–2
10–1
Ca
100
101
Figure 5. The fraction α of the tube’s cross-section that is occupied by the propagating finger
as a function of the capillary number Ca = µU/σ for tubes with square cross-sections. Results
for tubes of rectangular, circular and elliptical cross-sections are qualitatively similar.
large finger velocities, the resistance to the flow is dominated by the viscous losses
and the propagating finger will tend to open a single path through the branching
network.
Without the approximations used in the derivation of (4.6), we must determine the
signs of various terms in (4.4) and (4.5). References R show that dC(U )/dU > 0 and,
using Hazel & Heil’s (2002) computational results, we find that α + U (dα/dU )|U > 0
(see figure 5 for the case of tubes with square cross-section; the curves for tubes
with rectangular, circular and elliptical cross sections are qualitatively similar). Hence
G(U, t) > 0, implying that the growth or decay of the perturbations is determined by
the sign of F which we rewrite in non-dimensional form as
K
dα(Ca)
K
F(Ca) = α(Ca) 2 −
+ Ca
1−
.
(4.7)
Ca
dCa
Ca
Here, Ca = µU/σ is the capillary number based on the propagation speed of the
symmetrically propagating fingers and the dimensionless parameter K = k/(Rσ ) is a
measure of the stiffness of the elastic end chambers.
Figure 6 shows plots of F(Ca) for square tubes and for a range of values of the
parameter K. Also shown (as dashed lines) are the approximations F(Ca) ≈ 2 −
K/Ca which correspond to the simplifications α ≡ 1 and C ≡ 0 used in the derivation
of (4.6). Figure 6 shows that the critical velocity Uc obtained from the approximate
analysis provides an excellent prediction for critical velocity at which the symmetrically
branching solution becomes unstable. This is because the capillary pressure jump
affects only the magnitude of the growth rate, but not its sign, therefore setting C ≡ 0
does not affect the prediction for Uc . Furthermore, at small capillary number, the
thickness of the fluid film that the propagating finger deposits on the channel walls
is small, therefore the approximation α ≡ 1 becomes more accurate as the capillary
number is reduced. This explains why the discrepancy between the exact and the
approximate solutions for Uc decreases with Ca. Both analyses show that if k = 0,
corresponding to the case when the end chambers are open to the atmosphere and
292
2
1
= 10–3
10–2
Asymmetric
0
Symmetric
10–1
100
–1
10–3
10–2
10–1
Ca
100
101
Figure 6. The function F(Ca) for tubes of square cross-section, plotted for various values of
the stiffness parameter K. If F < 0 (> 0), the finger branches symmetrically (asymmetrically).
The dashed lines represent the approximation F = 2 − K/Ca which provides excellent
predictions for F(Ca) = 0. Results for tubes of rectangular, circular and elliptical cross-sections
are qualitatively similar.
offer no resistance to the flow, we have F > 0, implying that the finger will branch
asymmetrically.
5. Comparing theory and experiment
To compare theory and experiment, we must estimate the (volumetric) stiffness
k of the end chambers. As discussed in § 2, the end chambers are circular holes
of radius a = 1.6 mm which are sealed with elastic membranes. Before the start of
the experiment, the membranes were pre-stretched by injecting silicone oil until
≈ 400 µm. Eschenauer &
their centres were deflected outwards by approximately w
Schnell’s (1986) large-displacement analysis of pressure-loaded circular membranes
provides the relation between the chamber pressure and the membrane’s maximum
and the corresponding chamber volume, V (w).
Using these results
deflection, pelast (w),
we obtain the volumetric stiffness
−1
2
6Eh3 (23 − 9ν) w
8
dpelast dV
=
+
,
(5.1)
k=
dw
dw
πa 6
7(1 − ν) h
3(1 − ν 2 )
The volume of PFD injected during
which has a strong nonlinear dependence on w.
the actual experiment is small relative to the total volume of the pressure chambers.
Therefore, the injection of the PFD finger causes only small additional deflections and
k can be expected to remain approximately constant throughout the experiment. The
flow resistance R of a rectangular channel of width W and aspect ratio 2:1 is given
by R = 139.93/W 4 (see White 1991). Using the estimates for the physical parameters
given in § 2, we obtain K = k/(Rσ ) ≈ 1.0 × 10−3 , indicating that the transition to nonaxisymmetric branching should occur at a capillary number of Ca ≈ K/2 = 5.0 × 10−4 .
This compares favourably with the experimental data of figure 4 which shows that
the transition between symmetric and asymmetric branching occurs for a finger
293
velocity (in the parent branch) of U ≈ 2U between 478 µm s−1 and 606 µm s−1 . This
corresponds to capillary numbers (based on the velocities in the daughter branches)
between Ca = µU/σ = 4.78 × 10−4 and 6.06 × 10−4 .
6. Discussion
We have studied the propagation of low-viscosity fingers in a fluid-filled branching
network of microchannels and have analysed the behaviour of the finger as it
passes a single bifurcation. The behaviour was shown to depend mainly on the
relative importance of viscous and elastic forces, characterized by the dimensionless
parameter λ = k/(2µUR). Viscous effects dominate if the finger velocity and/or the
viscous flow resistance are large, so that λ < 1. In this regime, the finger branches
asymmetrically and will tend to open a single path through the network. This is
because any perturbation that increases the length of one air finger relative to the
length of the other, reduces the viscous flow resistance offered by the column of
viscous fluid ahead of the finger tip. This causes the velocity of the longer finger to
increase, enhancing the initial difference in finger length even further. We note that
the ‘inverse’ of this mechanism is responsible for the symmetric propagation of the
oil fingers that displace the air that initially fills the channel when the experiment is
first set up. If a small local imperfection in the channel geometry momentarily allows
one of the oil fingers to propagate slightly faster than the other, the resulting increase
in the finger length increases the flow resistance and causes the finger to slow down,
allowing the other finger to catch up.
Conversely, if the air finger propagates sufficiently slowly and/or if the stiffness
of the end chambers is sufficiently large so that λ > 1, the finger tends to branch
symmetrically. This is because the slightly stronger inflation of the elastic end chamber
that is connected to the branch conveying the longer finger, creates a strong restoring
pressure which reduces the finger velocity until the volumes of both end chambers
are approximately equal again.
In the context of the pulmonary airway reopening problem, our results suggest that
the reopening of occluded airways should be performed at small speeds to encourage
symmetric branching of the air finger as it propagates into the bronchial tree, though
it is important to re-iterate that our model provides an extreme simplification of the
conditions in the pulmonary airways. For instance, our model is based on a highly
idealized geometry; our lumped representation of the wall elasticity ignores the fact
that the wall deformation interacts with the fluid flow near the finger tip. Furthermore,
while symmetric branching of the propagating air finger is clearly desirable, any
attempt to optimize airway reopening procedures will be subject to many additional,
and possibly conflicting, constraints. For instance, Bilek, Dee & Gaver (2003) showed
that airway reopening at small flow rates may result in cellular damage owing to
an increase in the stresses on the airway wall. Finally, we assumed the finger to be
driven by an imposed flow rate rather than a controlled driving pressure. In § 4, we
showed that d
pdr /dt = 0, implying that there is no difference between the two cases,
within the framework of a linear stability analysis. However, nonlinear effects are
likely to become important when the deviations in the finger velocities become large.
It is therefore conceivable that changes to the driving mechanism could affect the
behaviour of the asymmetrically branching solution (see e.g. Halpern et al. 2005).
As for microfluidic flows, the results suggest that some limitations exist on the
ability to fill a network of channels evenly or to divide drops. For this reason, it will
be important to explore ways of stabilizing the symmetric branching in lab-on-a-chip
294
applications, for example, by introducing an increased resistance to the flow as the
finger advances, through the use of elastic forces.
M. H. was supported in part by an EGIDE visiting fellowship at Ecole Polytechnique. S. T. was supported in part by the MIT-France program. The authors acknowledge technical help from José Bico.
REFERENCES
Bilek, A. M, Dee, K. C. & Gaver, D. P. 2003 Mechanisms of surface-tension-induced epithelial
cell damage in a model of pulmonary airway reopening. J. Appl. Physiol. 94, 770–783.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.
Calderon, A. J. & Bull J. L. 2004 Homogeneity of bubble transport through a bifurcation for gas
embolotherapy. FASEB J. 18, A373.
Cassidy, K. J., Gavriely, N. & Grotberg, J. B. 2001 Liquid plug flow in straight and bifurcating
tubes. J. Biomech. Engng 123, 580–589.
Eschenauer, H. & Schnell, W. 1986 Elastizitätstheorie I . B. I.-Wissenschaftsverlag, Mannheim.
Gaver, D. P., Samsel, R. W. & Solway, J. 1990 Effects of surface tension and viscosity on airway
reopening. J. Appl. Physiol. 69, 74–85.
Halpern, D., Naire, S., Jensen, O. E. & Gaver, D. P. III 2005 Unsteady bubble motion in a
flexible-walled channel: predictions of a viscous stick–slip instability. J. Fluid Mech. 528,
53–86.
Hazel, A. L. & Heil, M. 2003 Three-dimensional airway reopening: the steady propagation of a
semi-infinite bubble into a buckled elastic tube. J. Fluid Mech. 478, 47–70.
Hazel, A. L. & Heil, M. 2002 The steady propagation of a semi-infinite bubble into a tube of
elliptical or rectangular cross-section. J. Fluid Mech. 470, 91–114.
Homsy, G. 1987 Viscous fingering in porous media. Annu Rev. Fluid Mech. 19, 271–311.
Link, D., Anna, S., Weitz, D. & Stone, H. 2004 Geometrically mediated breakup of drops in
microfluidic devices. Phys. Rev. Lett. 92, 054503.
Quake, S. R. & Scherer, A. 2000 From micro- to nano-fabrication with soft materials. Science
290, 1536–1540.
Reinelt, D. & Saffman, P. 1985 The penetration of a finger into a viscous fluid in a channel and
tube. SIAM J. Sci. Stat. Comput. 6, 542–561.
Song, H., Tice, J. & Ismagilov, R. 2003 A microfluidic system for controlling reaction networks in
time. Angew. Chem., Intl Edn 42.
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161–165.
White, F. 1991 Viscous Fluid Flow . McGraw-Hill.
Wong, H., Radke, C. J. & Morris, S. 1995 The motion of long bubbles in polygonal capillaries.
Part 1. Thin films. J. Fluid Mech. 292, 71–94.
Journal of Colloid and Interface Science 308 (2007) 231–238
www.elsevier.com/locate/jcis
Transport of wetting liquid plugs in bifurcating microfluidic channels
Cédric P. Ody, Charles N. Baroud ∗ , Emmanuel de Langre
Laboratoire d’Hydrodynamique (LadHyX), Ecole Polytechnique-CNRS, 91128 Palaiseau Cedex, France
Received 16 August 2006; accepted 5 December 2006
Available online 12 December 2006
Abstract
A plug of wetting liquid is driven at constant pressure through a bifurcation in a microchannel. For a plug advancing in a straight channel,
we find that the viscous dissipation in the bulk may be estimated using Poiseuille’s law while Bretherton and Tanner’s laws model the additional
dissipation occurring at the rear and front interfaces. At a second stage, we focus on the behavior of the plug flowing through a T-junction.
Experiments show the existence of a threshold pressure, below which the plug remains blocked at the entrance of the junction. Above this required
pressure, the plug enters the bifurcation and either ruptures or splits into two daughter plugs, depending on the applied pressure and on the initial
length of the plug. By means of geometrical arguments and the previously cited laws, we propose a global model to predict the transitions between
the three observed behaviors.
© 2007 Elsevier Inc. All rights reserved.
Keywords: Microfluidic bifurcation; Plug rupture; Threshold pressure; Splitting
1. Introduction
Microfluidic fabrication techniques provide new ways to
handle fluids in complex geometries at sub-millimeter scale.
While the transport of a single fluid is well understood in the
laminar “Stokes” regime, two-phase flows present important
challenges due to the nonlinear effects at the moving interfaces.
Indeed, even the transport of liquid plugs in a straight channel
involves a nonlinear pressure–velocity relationship that results
from the balance between viscous and capillary effects [1]. This
balance operates near the triple lines at which the liquid and
gas phases meet the solid substrate; the details of this balance
depend strongly on the wetting properties of the solid and the
liquid [2].
The transport of plugs in a complex geometry, such as a
network of channels, is of interest for microfluidics but also
in geological and biological situations. For instance, an understanding of the transport of plugs through bifurcations is necessary for applications in drug delivery in the pulmonary airway
* Corresponding author.
E-mail address: [email protected] (C.N. Baroud).
0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcis.2006.12.018
tree [3], or in the extraction of oil from porous rocks [4], where
one is interested in the transport of a volume of a liquid (mucus or petroleum) bounded by two liquid–gas interfaces. Since
the liquid usually wets the solid in these situations, this will
be represented at rest by a zero contact angle between the two
phases.
The aim of the present work is to study the transport of such
plugs in rectangular microchannels, when both inertia and gravity are negligible. This is done by driving plugs at constant
pressure through straight and bifurcating microchannels and
by developing corresponding theoretical models. A combined
theoretical and experimental study of the straight and bifurcating cases will therefore provide the essential building block for
understanding the dynamics of a two-phase flow in a channel
network, which in turn will allow the modeling of concrete situations.
The paper is structured as follows: Section 2 presents the
experimental setup and protocol. It is followed in Section 3 by
the development of a theoretical model for a plug in a straight
channel which is validated experimentally. Finally, Sections 4
and 5 provide the experimental and theoretical study of the flow
of a plug through a T-junction, respectively.
232
2. Experimental setup
A schematic of the experimental setup used in our study is
shown in Fig. 1. The experiments were conducted in rectangular
cross section microfluidic channels made of polydimethylsiloxane (PDMS) by using soft lithography techniques [5]. These
channels are integrated in a microfluidic chip, composed of two
parts. To obtain the upper part in which the channel is etched,
a thin layer of photosensitive resin (Microchem, SU8-2035) is
first spin-coated on a silicon wafer and patterned by standard
photolithography. The speed of spin-coating sets the thickness b
of the future channels, while the widths and the shapes are controlled by the design of the patterned masks which are used
during the UV exposure. The photoresist is then developed
yielding the mold on which a thick layer of PDMS (Dow Corming, Sylgard 184) is poured and allowed to partially cure. The
resulting cast is finally removed from the mold and inlet/outlet
holes are punched for later connections. In parallel, the lower
part of the microchannel is obtained by spin-coating and partially curing a flat layer of PDMS on a glass microscope slide.
At a final stage, the two PDMS parts are brought into contact
and the whole is cured completely, thus forming a microfluidic chip. As seen in Fig. 1, the microchannel consists of an
entrance region (Y-junction) used to form the liquid plugs, an
initial straight channel and a final T-junction. We note that wi
and wo are the widths of the inlet and outlet branches of the
T-junction. Typical values for the thickness and the width of
the straight channel were b 25–50 µm and wi 200–300 µm.
The different dimensions were measured on the resin mold with
a profilometer (Dektak).
A constant driving pressure Pdr was applied using a water
column connected to a sealed air bottle as shown in Fig. 1;
the air flowing out of the bottle was then driven to one inlet of the Y-junction while the other inlet was used for liquid introduction. Outlets of the T-junction were at atmospheric
pressure P0 . Different values of the driving pressure Pdr were
obtained by varying the height of water in the column and cal-
Fig. 1. Experimental setup. A constant pressure Pdr is applied at the upper
left inlet branch of the Y-junction. The lower left branch is used to introduce
the wetting liquid from a syringe pump. The motion of the plug is recorded
with a camera through a microscope. Note the enlargement of the inlet channel
downstream of the Y-junction.
Fig. 2. A liquid plug after its formation within the Y-junction. The plug moves
from left to right. The upper branch is connected to the constant pressure source
(air) and the lower branch contains the wetting liquid which is injected using
a syringe pump. Length and speed are calculated from the measured positions
of the menisci at the advancing (front) and receding (rear) interfaces, detected
along the centerline of the channel.
culating the corresponding hydrostatic pressure, ranging from
100 Pa (1 cm of water) up to 600 Pa (60 cm of water). The
precision on the water height pressure was 2 mm, giving a maximum error of 20% for the lowest pressure value. All pressure
losses in the external tubing were neglected.
The plugs were formed with perfluorodecalin (PFD), a fluorinated liquid with dynamic viscosity η = 5 cP and surface
tension γ = 15 mN/m [6]. In addition to wetting the solid
substrate, the PFD also has the advantage of not swelling the
PDMS [7]. To obtain the typical plug shown in Fig. 2, the liquid was forced into one inlet of the Y-junction by actuating
a syringe pump for a given interval of time while a manual
valve cut off the air entrance from the pressure source. Once
the plug was formed, the syringe pump was switched off, and
the pressure was applied. By controlling the duration of the
syringe pump actuation, plugs of different lengths were obtained. However, since the minimal length of the plugs was
fixed by the characteristic size of the Y-junction, the channel
was enlarged downstream of the Y-junction to allow the study
of shorter plugs, resulting in lengths as small as the width wi of
the inlet channel. There was no upper limit for the size of the
plugs.
Experiments were recorded with a high speed camera (resolution 1024 × 256 pixels, 1 pixel for 10 µm, sampled at 10–
100 fps) through a microscope (Leica, MZ 16). For each image
of the sequences thus obtained, the positions (xf and xr ) of the
menisci at the advancing (front) and receding (rear) interfaces
of the traveling plug were manually located along the centerline of the channel. These measurements yielded the length
and the speed of the plug during its transport as, respectively,
Lplug = xf − xr and Uplug = dxc /dt with xc = (xf + xr )/2. Note
that before starting data acquisition, several plugs of PFD were
introduced in order to prewet the channel walls.
A sketch of a plug moving in a rectangular microchannel is
shown in Fig. 3. In this schematic, plane A (“in-plane” section)
represents the projected view as seen through the microscope,
e.g., Fig. 2. We will also refer later to plane B as the “inthickness” section.
233
Fig. 4. Sketch of the plug (sections A and B). Here, θ a is the dynamic contact
angle at the front interface and eb (ewi ) is the thickness of the deposited film at
the rear of the plug in the thickness (width).
Fig. 3. Schematic of a plug in a rectangular channel. Planes A and B correspond
respectively to “in-plane” and “in-thickness” sections.
3. Visco-capillary regime in straight channels
We first consider the transport of a liquid plug of length L0
moving with a steady velocity U0 in the straight microchannel
upstream of the T-junction. By taking a characteristic length
scale D ∼ 50 µm and typical velocities U0 ∼ 5 mm/s, one obtains a Bond number Bo = ρgD 2 /γ ∼ 5×10−3 and a Reynolds
number Re = ρU0 D/η ∼ 2 × 10−2 , where ρ ∼ 1000 kg/m3
and g = 9.81 m/s2 are, respectively, the density of the liquid
and the acceleration of gravity. Therefore, one may neglect
gravity and inertial effects, expecting the plug dynamics to be
governed by a visco-capillary regime as mentioned in [8]. Here,
we use an approach similar to the one developed by Bico and
Quéré [1]. Given a constant driving pressure Pdr , the steadystate pressure balance across the plug is
r
a
Pdr = Pcap
+ Pvisc + Pcap
,
r
Pcap
(1)
a
Pcap
where
and
express the capillary pressure drops at
the receding and advancing interfaces of the plug, respectively,
while Pvisc represents the viscous dissipation occurring in the
bulk. Using Poiseuille’s law, the latter is expressed as
Pvisc = αηL0 U0 ,
(2)
with α, a dimensional coefficient, corresponding to the geometry of the channel. For a rectangular geometry [9], α may be
approximated as
25 b −1
12
.
α 2 1−6 5
(3)
b
π wi
r and P a
When the plug is at rest, static values of Pcap
cap
r
a
are given by Laplace’s law as Pcap = −Pcap = γ κ, where
κ 2(b−1 + wi−1 ) is the mean curvature of each interface at
rest. Here, we neglect the flow of liquid along the corners of the
channel that slightly deforms the shape of the interfaces [10].
When the plug is moving, the front and rear curvatures are
modified as shown in Fig. 4. At the front interface, the balance
between friction and wetting forces at the vicinity of the contact
line leads to the existence of a non-zero dynamic contact angle
and this flattening of the advancing interface increases the resistance to the motion. By assuming that θ a has the same value
in the two sections, the dynamic capillary pressure drop for the
front interface becomes
a
Pcap
= −γ κ cos θ a .
(4)
The relation between the contact angle and the velocity of the
contact line is known through Tanner’s law [1,11] which states
that
θ a = (6Γ Ca)1/3 ,
(5)
where Ca = ηU0 /γ is the capillary number and Γ is a logarithmic prefactor that accounts for the singularity occurring at
the contact line, where the usual no-slip boundary condition at
the solid surface leads to a logarithmic divergence in the shear
stress [12].
Using Eq. (5) and taking the first-order term in the Taylor
expansion of cos θ a in Eq. (4), the front pressure drop becomes
(6Γ Ca)2/3
a
.
Pcap γ κ −1 +
(6)
2
At the rear interface, a thin film is deposited at the channel
walls [13], which reduces the rear meniscus radius and thus,
also increases the resistance to the motion. We note that eb and
ewi are the thicknesses of the deposited film at the rear of the
plug in the in-thickness and in-plane sections, respectively. The
dynamic capillary pressure drop at the rear interface may then
be written as
1
1
r
=γ
+
.
Pcap
(7)
b/2 − eb wi /2 − ewi
Bretherton’s law expresses the thickness e of the deposited film
at the rear of a plug as a function of the capillary number Ca
and the radius R of a circular capillary tube [1]:
e/R = 3.88Ca2/3 .
(8)
Assuming Bretherton’s law to be valid in both directions of the
rectangular cross section, i.e., 2eb /b = 2ewi /wi = 3.88Ca2/3 ,
the rear capillary jump is obtained from Eq. (7) as
r
= γ κ 1 + 3.88Ca2/3 .
Pcap
(9)
Finally, by using Eqs. (2), (6) and (9), one derives
Pdr αηL0 U0 + γ κβCa2/3 ,
(10)
is a nondimensional coefficient
where β = 3.88 +
obtained from Bretherton’s and Tanner’s laws.
Equation (10) governs the dynamics of a plug of length L0
moving steadily at speed U0 in a straight rectangular channel
for a constant driving pressure. Equation (10) can be nondimensionalized by comparing it with the capillary pressure jump γ κ,
and we obtain
(6Γ )2/3 /2
Pdr αCa + βCa2/3 ,
(11)
234
Fig. 5. Capillary number of a liquid plug as a function of its inverse nondimensional length 1/α for a pressure Pdr = 250 Pa (Pdr = 0.2). The dashed line
corresponds to Poiseuille’s law, Ca ∝ 1/α. The full line corresponds to Eq. (11)
with β = 16.
with Pdr = Pdr /(γ κ) is the nondimensional driving pressure
and α = αL0 /κ is the nondimensional length.
Experiments were conducted in a microchannel to verify
the applicability of Eq. (11) to our rectangular geometry. The
cross section of the microchannels used here was wi × b =
210 × 25 µm. Single plugs of various lengths were introduced
and forced at constant pressure following the protocol of Section 2. For all driving pressures, we observed that the measured
lengths and speeds of the plugs remained constant within 5%
throughout the channel and average length L0 and speed U0
were determined.
A sample of experimental results is plotted in Fig. 5. Data
corresponding to long plugs asymptote to Poiseuille’s law
(dashed line, Pdr = αCa from Eq. (2)). For shorter plugs, the
data exhibit a nonlinear Ca vs (α)−1 relationship due to an additional dissipation occurring at the interfaces. The theoretical
curve (full line) obtained from Eq. (11) is plotted with β = 16
and is in good agreement with the experimental data. The corresponding value of Γ = 20 is higher than experimental values
reported in millimetric circular tubes [1]. However, the coefficient (6Γ )1/3 = 4.9 appearing in Tanner’s law (Eq. (5)) is in
close agreement with Hoffman’s data [11] for which the previous coefficient is of order 4–5 for a dry tube. A more precise estimate of the Tanner’s constant would require a specific study by
keeping track of both the microscopic and macroscopic physics
involved in our configuration, as described in [12].
Thus, although Bretherton’s and Tanner’s laws are not
strictly applicable to the rectangular geometry, our results show
the possibility of using the previous laws to model the dynamics
of plugs in our channels.
4. Blocking, rupture or splitting at the T-junction
We now consider a plug flowing through the T-junction for
different driving pressures and initial plug lengths, L0 , measured in the straight channel upstream of the bifurcation. Exper-
Fig. 6. Blocking of the plug at the entrance of the T-junction for two distinct
pressures with Pdr < Pthresh . The front interface is pinned at the corners of the
bifurcation and adapts its curvature to the applied pressure. The shape of the
rear interface is independent of Pdr .
iments were conducted with channel dimensions wi × wo × b =
260 × 260 × 46 µm. Three different behaviors were observed
depending on the driving pressure and on the initial length of
the plug as described below.
4.1. Blocking
The first feature extracted from the observations of a plug
getting to the T-junction is the existence of a threshold pressure Pthresh (between 200 and 300 Pa for our experiments)
which does not depend on the initial length of the plug. For
Pdr < Pthresh , the plug remains blocked at the entrance of the
bifurcation as presented in Fig. 6, which shows the equilibrium positions of plugs for two distinct pressures (Pdr = 100,
200 Pa). While the rear interface has the same shape in both
cases, the front interface adapts its curvature to balance the applied pressure keeping its extremities pinned at the corners of
the bifurcation.
4.2. Rupture
When the pressure exceeds the threshold pressure Pthresh , the
plug continues its propagation through the bifurcation. If the
plug is short, its rear interface catches up with its front interface
before the latter reaches the wall opposite to the entrance channel. The plug then ruptures, opening the outlet branches of the
T-junction to air. A typical sequence of plug rupture is shown in
Fig. 7. Eventually, the liquid that remains on the channel walls
drains slowly through the action of capillary forces and air drag.
4.3. Splitting into two daughter plugs
When the plugs are sufficiently long that the front interface
reaches the opposite wall, two daughter plugs of equal length
are formed through a splitting of the initial one, as shown in
Fig. 8. As soon as the liquid reaches the wall, the transport
235
Fig. 7. Sequence of plug rupture. The plug ruptures in the T-junction, leaving liquid on one side of the outlet channel, where air can flow freely from the pressure
source.
dynamics becomes dominated by the wetting forces acting between the two phases. These forces act to draw the liquid very
rapidly into contact with the wall, as observed between the
images (b) and (c) of Fig. 8. In part (c), we see that the curvature is not constant along the liquid surface, implying local
pressure gradients in the fluid which pull it into the daughter
branches. These wetting forces are dominant compared to the
driving pressure, as seen by the rapidity of the advance once
the plug has touched the wall.
Note that before rupture or splitting, the evolution of the plug
in the bifurcation is similar in both cases: the front interface
advances without constraints while the extremities of the rear
interface remain pinned by the corners of the bifurcation until
process of rupture or splitting is achieved (Fig. 7c).
4.4. Phase diagram
Fig. 9. Phase diagram for plug behavior at the T-junction.
We summarize the three cases discussed above (blocking,
rupture and splitting) in the experimental phase diagram shown
in Fig. 9. As stated above, we observed the blocking () of the
plug below a threshold pressure, Pthresh , whose value is seen to
be independent on the length of the plug. Above Pthresh , experimental data show that the transition between rupture () and
splitting (!) depends on the applied pressure and on the length
of the plug: as the pressure increases, the length of liquid required to create two daughter plugs becomes smaller.
In this section, the two transitions appearing in the phase diagram of Fig. 9 are explained theoretically. We first establish a
2D pressure balance to derive the value of the blocking pressure after which we develop a model to predict the transition
between rupture and splitting.
5.1. Threshold pressure
Since the plug is at rest in the blocking case, the viscous term
in Eq. (1) is zero and the new pressure balance may be written
as
r
a
+ Pcap
.
Pdr = Pcap
(12)
r and P a express static pressure drops
Here, the terms Pcap
cap
and Eq. (12) may be simplified by distinguishing in-plane and
236
Fig. 10. The shapes of the rear and front interfaces at the threshold pressure
(in-plane section).
in-thickness contributions of the capillary pressure jumps which
are related via Laplace’s law to the corresponding principal curvatures of the interfaces. Since the channel thickness does not
vary at the bifurcation, we assume that the in-thickness components of the pressure drops at the rear and front interfaces
cancel each other, the radii of curvature in the thickness being
b/2 in absolute value. Therefore, the problem is reduced to a
2D pressure balance of in-plane components of pressure drops,
leading to
1
1
Pdr = γ
(13)
+
,
Rr Ra
where R r and R a are the radii of curvature of the rear and front
interfaces in plane A, respectively.
The experimental observations shown in Fig. 6 provide the
information to derive the value of the threshold pressure. For
Pdr < Pthresh , the receding interface is seen to keep a constant
in-plane radius of curvature R r = wi /2 while the advancing
interface adapts its in-plane curvature to the applied pressure
and its radius R a varies such that |R a | wi /2. The situation
in which R a = +wi /2 is illustrated in Fig. 10 and leads to the
threshold pressure, which is the highest pressure that can be
sustained across the plug, reading finally
Pthresh =
4γ
.
wi
(14)
For our experimental conditions (wi = 260 µm and γ =
15 mN/m), we obtain Pthresh 230 Pa, in agreement with the
measured value (200 Pa < Pthresh < 300 Pa).
5.2. Rupture or splitting
We now propose to explain the critical plug length, Lcrit ,
for getting rupture or splitting when the pressure exceeds the
threshold pressure. Our aim here is not to model the complete
dynamics of the plug but to derive a global model which clarifies the dependence of Lcrit on the driving pressure Pdr by
taking into account the fundamental mechanisms involved in
the plug evolution. We first consider a plug of length Lcrit theoretically situated on the transition curve. A geometric condition
is then obtained from the shape of such a plug in the T-junction.
Finally, a dynamic argument is used to map this geometric condition onto the experimental L0 –Pdr diagram.
(a)
(b)
(c)
Fig. 11. Shape of the plugs before (a) rupture and (c) splitting. (b) Schematic
of a plug of volume Vcrit situated on the theoretical boundary between rupture
and splitting where front and rear interfaces of a plug meet at the wall.
5.2.1. Geometric condition
The diagram of Fig. 9 exhibits a L0 –Pdr dependent boundary
between the rupture or splitting cases. We consider the shape of
the plugs just before rupture or splitting, as shown in Figs. 11a
and 11c. In Fig. 11a, the rear interface catches up with the front
interface before the latter reaches the wall, leading to rupture
of the plug. Conversely, in Fig. 11c, the front interface is about
to touch the wall before the rear interface catches up with it,
resulting in the splitting of the plug and the formation of two
daughter plugs. In the limiting case, a plug of length Lcrit sees
its front and rear interfaces meeting at the wall. Approximating
the shape of both interfaces with circular sections leads to the
sketch of Fig. 11b. Here, the critical volume of the plug is noted
Vcrit and is completely defined by θ a , θ r , wo and the thickness b
of the microchannel. At point C, the rear contact angle θ r is defined geometrically as θ r = 2 arctan(wi /2wo ). In contrast, the
contact angle θ a at point E may take different values depending on the velocity of the advancing interface. As a result, the
critical volume decreases as θ a increases for a given T-junction,
down to a zero volume when θ a = π − θ r .
The volume of liquid contained between the two interfaces in
the critical configuration corresponding to Fig. 11b is obtained
as
Vcrit = bwo2 f θ a − g θ r ,
(15)
where
−2
f θ a = θ a − cos θ a sin θ a 1 − cos θ a
(16)
and
−2
g θ r = π − θ r + cos θ r sin θ r 1 + cos θ r .
(17)
This volume corresponds to a plug of length Lcrit in the entrance
channel such that
Vcrit = b Lcrit wi + (1 − π/4)wi2 ,
(18)
with the correction term b(1 − π/4)wi2 taking into account the
volume contained in the circular menisci. Therefore, the critical
length is such that
2
r wo a π
Lcrit
=
f θ −g θ − 1−
.
wi
wi
4
237
(19)
For a given contact angle θ a , a plug of initial length L0 < Lcrit
will rupture in the bifurcation while a plug such that L0 > Lcrit
will produce two daughter plugs.
Since θ r is a function of the ratio wo /wi , the value of Lcrit
depends on θ a and wo /wi . Therefore, Eq. (19) will provide different values of Lcrit as θ a varies, the other quantities being
constant for a given geometry. The problem is now to understand the mechanisms involved in the determination of θ a .
5.2.2. Dynamic condition
Although the propagation in the bifurcation is unsteady, the
dynamics is still governed by a visco-capillary regime. Therefore, at a given time, we may use Eq. (1) on a specific fluid
streamline. For simplicity, we choose the streamline linking
points C and E.
Close to point C, the speed of the rear interface is nil since it
is pinned at the corners of the junction. Thus, the capillary drop
at the receding interface is approximated as
1
2
r
=γ
+
,
Pcap
(20)
Rr b
where R r is the in-plane radius of the receding interface (see
Fig. 11b) and is expressed as
wo
.
Rr =
(21)
1 + cos θ r
For the front interface, we assume that the speed of the contact
line at point E establishes the contact angle θ a and that θ a is the
same in the thickness and in the plane. Therefore, the capillary
drop at the advancing interface reads
1
2 cos θ a
a
−
,
Pcap = γ
(22)
Ra
b
where R a is the in-plane radius of the advancing interface, calculated from geometry as
wo
.
Ra =
(23)
1 − cos θ a
The viscous term in Eq. (1) is approximated as a dissipation
term in a Hele–Shaw geometry (since b/wo 0.18), leading to
12η
Lm Vm ,
(24)
b2
where Lm is the length of the streamline C–E and Vm the mean
speed of the fluid particles between points C and E. As stated
above, the speed at point C is nil and the speed at point E is
related to the contact angle by Tanner’s law (Eq. (5)) as
Pvisc =
VE =
γ
γ (θ a )3
CaE =
.
η
6ηΓ
(25)
We estimate the quantities Vm and Lm to first order as Vm =
(VE + VC )/2 and Lm = yE − wi /2, where yE = wo / tan(θ a /2)
is the distance from point E to the centerline of the inlet channel
(see Fig. 11b).
Fig. 12. Experimental Pdr –L0 /wi diagram and theoretical transitions. (1) Rupture-splitting transition corresponding to Eq. (27) with θ a varying from π/6
to π/3 and (6Γ )1/3 = 4.9. (2) Threshold pressure (Eq. (14)).
Finally, combining the terms above leads to the pressure balance across the plug
yE − wi /2 a 3
1
1
2
a
Pdr γ 2
+ a + r + 1 − cos θ ,
θ
R
R
b
Γ b2
(26)
which accounts for the dynamic relation between θ a and Pdr .
Using a reference pressure γ /wi and expressing the different
quantities as functions of θ a , wi /wo , wo /b and wi /b, Eq. (26)
reads in a nondimensional form
2 a 3
wo w i
wi
θ a −1
(θ )
Pdr 2
−
tan
b b
2
b
Γ
wi
wi
+ 1 + cos 2 arctan
2wo
wo
2wi
wi +
(27)
+
1 − cos θ a .
b
wo
5.2.3. Summary of the model
The geometrical condition provides a relation between Lcrit
and θ a (Eq. (19)) while θ a may be related to Pdr using the
pressure balance of Eq. (26). For a given value of θ a , we are
now able to estimate the value of Pdr corresponding to Lcrit . By
varying θ a from π/6 to π/3, we obtained the results plotted in
their dimensionless form in Fig. 12, using Γ = 20 which is the
value obtained from straight channel experiments. The quantitative and qualitative behaviors of the theoretical transition are
in good agreement with the experimental data.
6. Conclusion
Our experiments on plug propagation in straight microchannels of rectangular cross section showed good agreement with
previous studies in capillaries with circular sections [1], in spite
of the large aspect ratio of our channels. In particular, the two
principal radii of curvature could be used to account for the
capillary effects at the advancing and receding interfaces and
238
effects of draining in the corners could be ignored for sufficiently fast propagation of the plugs.
Experiments through bifurcations show three distinct behavior regimes: blocking of the plugs at the bifurcation for small
driving pressures, rupture and channel reopening, or splitting
into two daughter plugs. The transitions between these three
behaviors may be explained by invoking geometric constraints
combined with capillary effects for the case of blocking, or a
visco-capillary balance for the rupture vs splitting transition.
These results reproduce some of the known phenomenologies in pulmonary airway flows and geological flows, such as
the existence of threshold pressures [10,14,15] or the different
mechanisms of transport of a liquid bolus [16,17]. However,
they benefit from having a well described geometry and direct
access to the complete plug dynamics at all times. Combined
with the advantages of microfluidic fabrication techniques, this
study will lead to new experimental models of network dynamics, with implications in biofluid or porous media flows.
Acknowledgment
The authors acknowledge helpful discussions with José Bico
and Matthias Heil.
References
[1] J. Bico, D. Quéré, J. Colloid Interface Sci. 243 (2001) 262–264.
[2] P.G. de Gennes, Rev. Mod. Phys. 57 (1985) 827–863.
[3] D. Halpern, O.E. Jensen, J.B. Grotberg, J. Appl. Physiol. 85 (1998) 333–
352.
[4] R. Lenormand, C. Zarcone, SPE Proc. (1984) 13264.
[5] S.R. Quake, A. Scherer, Science 290 (2000) 1536–1540.
[6] H. Song, J.D. Tice, R.F. Ismagilov, Angew. Chem. Int. Ed. 42 (2003) 768–
772.
[7] J.N. Lee, C. Park, G.M. Whitesides, Anal. Chem. 75 (2003) 6544–6554.
[8] P. Aussillous, D. Quéré, Phys. Fluids 12 (2000) 2367–2371.
[9] H.A. Stone, A.D. Stroock, A. Ajdari, Annu. Rev. Fluid Mech. 36 (2004)
381–411.
[10] R. Lenormand, J. Fluid Mech. 135 (1983) 337–353.
[11] R.L. Hoffman, J. Colloid Interface Sci. 50 (1975) 228–241.
[12] C. Treviño, C. Ferro-Fontán, F. Méndez, Phys. Rev. E 58 (1998) 4478–
4484.
[13] F.P. Bretherton, J. Fluid Mech. 10 (1961) 166.
[14] A. Majumdar, A.M. Alencar, S.G. Buldyrev, Z. Hantos, Z.H. Stanley,
B. Suki, Phys. Rev. E 67 (2003) 031912.
[15] B. Legait, C. R. Acad. Sci. Paris, Sér. II 292 (1981) 1111–1114.
[16] F.F. Espinosa, R.D. Kamm, J. Appl. Physiol. 86 (1999) 391–410.
[17] K.J. Cassidy, N. Gavriely, J.B. Grotberg, J. Biomech. Eng. 123 (2001)
580–590.
Chapter 3
Droplet manipulation through
localized heating
69
The work presented in this chapter is the result of several years of research
and many collaborations. It is not presented in chronological order. The external collaborators who participated on this work were Jean-Pierre Delville,
Matthieu Robert de Saint Vincent, Régis Wunenburger (all from U. Bordeaux I); François Gallaire (U. Nice); David McGloin and Dan Burnham (U.
Saint Andrews). At LadHyX, the quantitative experiments were performed
by Maria-Luisa Cordero and Emilie Verneuil.
3.1
Manipulating drops with a focused laser
We now focus on the manipulation of droplets in microfluidic circuits. Two guiding principles seemed important in developing manipulation techniques which lead to our early
experimenting with lasers in microchannels. The first principle was to take advantage
of scaling laws favorable to miniaturization such as by using surface phenomena, rather
than body forces which become weak at small scales. Second, our design strategy was
to simplify the microfabrication and minimize the cost of the channels, which is achieved
by fabricating the simplest PDMS channels and not requiring electrode deposition nor
multi-layer lithography.
With these general principles in mind, the approach that we converged upon was the
use of a laser to induce Marangoni flows on the surface of droplets, thus satisfying the two
above criteria. Furthermore, since optical manipulation techniques are well developed,
namely for the use with optical tweezers [59], the approach seemed technically feasible,
even though it was rather risky. Indeed, no one had demonstrated the manipulation of
microfluidic drops larger than a few microns with lasers at the time when our project was
initially proposed (and funded). Optical forces, which generate less than nanoNewtons,
seemed too weak to be used as a backup in case the Marangoni effect was not obtained.
The first effect that was observed, once we applied laser-heating to a water-oil
interface, is shown in Fig. 3.1. This figure shows a cross-shaped microchannel in which
oil is forced from the top and bottom channels and water from the left channel. In the
absence of laser forcing, water drops in oil are produced in a regular fashion, much like the
flow-focusing devices that have become standard. When the laser is illuminated, however,
the water-oil interface is completely blocked for times much longer than a typical drop
detachment time. In the case shown here, the water is forced at a constant flowrate such
that the neck connecting the drop to the water input grows as the drop is held in place,
meaning that the drop that finally detaches is larger than the drop produced without the
laser action.
The time that the drop is blocked depends on the laser power and its location,
though the blocking works equally well for any drop formation geometry. This is shown
in Fig. 3.2, where the blocking time is measured as a function of laser position and power,
for a cross and T geometries. We observe a linear increase in blocking time with laser
power in both cases, with a threshold value below which the drops are not held. Finally,
we also observe that the drop size increases linearly with holding time, at least for certain
blocking positions for an imposed flowrate.
Note however that the experiments of Figs. 3.1 and 3.2 were all performed using
a visible laser (Argon Ion, wavelength λ = 514 nm). In this case, heating the water is
70
Laser off
Laser on
(a)
(b)
(c)
(d)
(e)
(f)
Laser
Figure 3.1: Microfluidic valve: In the absence of a laser, water drops are produced in a
regular fashion. When the laser is illuminated, the shedding is delayed and the drop that
finally forms is larger.
only possible if a dye is dissolved in it; this set of experiments was done using fluorescein
because it absorbs at 514 nm. For this reason, it is not clear if the blocking time is
determined by intrinsic effects in the drops or by photo-bleaching of the sensitive fluorescein molecules. In later experiments where the water was flowed at constant pressure
and where an infrared laser was used with no dye in the water, we were able to achieve
blocking times larger than one minute in a true flow-focusing device, although systematic
studies were not performed.
These initial experiments seemed extremely promising from a practical point of view
since they represented (and still represent) the only available method for modulating the
shedding frequency and size of drops, without the need to vary the external flow controls.
This lead the CNRS and Ecole Polytechnique to deposit a patent on the manipulation
(a)
(b)
Figure 3.2: Characterization of the blocking time as a function of laser position and
power, for two different geometries, using an Argon Ion laser. Note that the phenomenon
does not depend strongly on the geometry but the blocking time increases with laser
power.
71
technique of drops and microfluidic systems through laser heating. At the same time,
the observed effects also raised a large number of questions, many of which remain open
at the time of writing of this memoir, although some advance has been made on certain
fronts and will be described below. The next sections describe the theoretical models
we have developed and the detailed experimental measurements which allow us to make
predictions on the limits of the technique. These are followed, in Section 3.4, by some
examples of what can be done with this approach, first with a simple Gaussian laser
spot, then using more complex beam-shaping and dynamic holographic techniques in
Section 3.5.
3.2
How does it work?
The initial observations were difficult to understand since the physical origin of the force
holding the drop were unknown. Indeed, the classical Marangoni forcing, as described
for example by Young et al. [124], is shown in Fig. 3.3. In this standard picture, the
surface tension decreases at the hot region relative to the cold region, which leads to a
flow along the surface of the drop that is directed from the hot to the cold side. This
drives the outer fluid to travel towards the cold side and the drop to migrate up the
temperature gradient, by reaction. This forcing has been demonstrated by experiments
and simulations [124, 88, 75] and it was not clear to us how the Marangoni effect could
generate a pushing rather than a pulling force on the interface. Moreover, optical forces
were ruled out for being too weak to trap a drop with a radius of a few hundred microns.
Marangoni
flow
COLD
drop motion
Induced external
flow
HOT
Figure 3.3: Classical view of a “swimming drop” which is subject to a temperature gradient. In the classical picture, drops migrate to the hot side.
However, the puzzle begins to clarify when particles are suspended in the fluids in
order to trace the flows, as shown in Fig. 3.4. Surprisingly, the flows are directed towards
the hot region, indicating an anomalous Marangoni effect which causes an increase in
surface tension with temperature. Owing to the linearity of the Stokes equations, one
therefore expects that changing the sign of the Marangoni flow will lead to changing the
sign of the force on the drop. This indicates that the origin of the repulsive force on the
drop may be of Marangoni origin, given that the flows are directed towards the hot region
along the interface.
The hydrodynamic model that was developed assumes a circular water drop placed
in a narrow gap with infinite lateral extent. This is similar to the Hele-Shaw model,
although we will look for the effects of shear on the drop, meaning that we will look for
72
Laser
Figure 3.4: Streaklines, inside and outside the drop, of the flows induced by the laser
heating. The arrows show that the flows are directed towards the hot regions along the
interface, indicating an anomalous Marangoni effect.
departures from potential flow theory. The basic image we would like to understand is
sketched in Fig. 3.5, which shows a drop submitted to an external flow going from the
left to the right. If the laser is able to stop the drop, the net effect due to the heating
needs to exactly counterbalance the drag force felt due to the external flow.
Surface tension imbalance
external
flow V
Force due
to surface
tension
drop
laser
Drag force
a
Figure 3.5: A drop is submitted to an external flow from left to right. The net effect due
to the Marangoni effect balances the drag due to this flow.
An estimate of the necessary force can therefore be obtained by estimating the drag
force on a drop in a narrow gap. This can be obtained from Ref. [90], where the force
acting on the drop is given as
πµoil R2
h
2λ
Fdrag '
1+λ+
V,
(3.1)
h
4R 1 + λ
where λ = µwater /µoil is the viscosity ratio, R is the drop radius, h is the gap height,
and V is the velocity of the oil at infinity. A numerical application of typical values
R = 100 µm, h = 10 µm, and V = 1 mm/s, yields a drag force of about 1 µN. This
force is many orders of magnitude larger than forces produced by optical techniques, thus
completely ruling out the possibility of an optical gradient origin of the force. It is also
several orders of magnitude larger than forces produced by dielectrophoresis [5].
The force generated by the thermocapillary convective flow on a droplet is investigated through the depth-averaged Stokes equations, since our channels have a large
width/height aspect ratio. We limit ourselves to the main features of the flow, the detailed derivations being left as an exercise to the reader [21, 29]: a circular drop of radius
73
!
"#$
"
%#$
%
!%#$
!"
!"#$
!!
!!
!"#$
!"
!%#$
&
(
!
%
'
%#$
"
"#$
!
Figure 3.6: Streamfunction contours obtained from the depth-averaged model described
in the text. Blue and red contours indicate counterclockwise and clockwise flows, respectively.
R is considered in an infinite domain and the flow due to the Marangoni stresses is
evaluated. Assuming a parabolic profile in the small dimension (h) and introducing a
streamfunction for the mean velocities in the plane of the channel, the depth averaged
equations, valid in each fluid, are
1 ∂2
1 ∂2
1 ∂ ∂
1 ∂ ∂
12
−
ψ = 0,
r +
r +
(3.2)
r ∂r ∂r r2 ∂θ2
r ∂r ∂r r2 ∂θ2 h2
where the depth-averaged velocities may be retrieved from uθ = −∂ψ/∂r and ur =
1/r(∂ψ/∂θ). The kinematic boundary conditions at the drop interface (r = R) are
zero normal velocity and the continuity of the tangential velocity. The normal dynamic
boundary condition is not imposed since the interface is assumed to remain circular, which
is consistent with our experimental observations, e.g. Fig. 3.4. Finally, the tangential
dynamic boundary condition, which accounts for the optically-induced Marangoni stress,
is
∂ u1θ
∂ u2θ
γ 0 dT
(3.3)
− µ2 r
=−
,
µ1 r
∂r r
∂r r
r dθ
are the velocities in the drop and the
where µ1,2 are the dynamic viscosities and u1,2
θ
0
carrier fluid, respectively. γ = ∂γ/∂T is the surface tension to temperature gradient,
which is positive in our case.
For simplicity, we approximate the steady state temperature distribution using a
Gaussian form T (x, y) = ∆T exp[−((x − R)2 + y 2 )/w2 ], where ∆T is the maximum
temperature difference between the hot spot and the far field and w corresponds to the size
of the diffused hot spot, which is significantly larger than the laser waist ω0 (see Sec. 3.3).
The equations are nondimensionalized using ∆T as temperature scale, R as length scale,
1 +µ2 )
Rγ 0 ∆T as force scale and R(µ
as time scale, the remaining nondimensional groups
γ 0 ∆T
being the aspect ratio h/R, the nondimensional hot spot size w/R and the viscosity ratio
µ̄ = µ2 /(µ1 + µ2 ).
A typical predicted flow field solving the above numerical formulation is shown in
Fig. 3.6, in which the four recirculation regions are clearly visible. The velocity gradients
display a separation of scales in the normal and tangential directions, as observed from
74
tangential shear stress
! r" = #
1 dru
r dr
v
! rr =2 #
dv
u
Total force
normal shear stress
Tangential
Shear Stress
Pressure
dr
Normal Shear
Stress
pressure
Figure 3.7: Left figure: Definition of the three stress fields that contribute to applying
a force on the drop. Right figure: Projections of the three fields along the x direction,
along with their sum. The solid black circle represents the zero level.
the distance between the streamlines in the two directions. Indeed, it may be verified
that the velocities vary over a typical length scale h/R in the normal direction, while the
tangential length scale is given by w/R.
Along with this flow field, we compute the pressure field, as well as the normal
(σ̄r̄r̄ = 2µ̄∂ ūr̄ /∂r̄) and tangential (σ̄r̄θ = µ̄ (1/r̄∂ ūr̄ /∂θ + ∂ ūθ /∂r̄ − ūθ /r̄)) viscous shear
stresses in the external flow, defined in Fig. 3.7(left). These projections on the x axis,
shown in Fig. 3.7(right), are then summed and integrated along θ to yield the total
dimensionless force (F̄ ) on the drop. Note that the global x component of the force is
negative and therefore opposes the transport of the drop by the external flow. The y
component vanishes by symmetry and the integral of the wall friction may be shown to
be zero since the drop is stationary.
Numerically computed values of F̄ R/h are shown by the isolated points in Fig. 3.8
as a function of w/R, for different values of the aspect ratio h/R. The points all collapse
on a single master curve, displaying a nondimensional scaling law F̄ ∝ wh/R2 , for small
w/R.
The dimensional form of the force can be obtained from scaling arguments, for small
h/R and w/R, by considering the three contributions separately and noting that the
velocity scale in this problem is imposed by the Marangoni stress. Using the separation
of scales along the azimuthal and radial directions, Eq. 3.3 becomes
(µ1 + µ2 )
U
∆T γ 0 R
∼
,
h
R w
(3.4)
where the ’∼’ is understood as an order-of-magnitude scaling. This yields the characteristic tangential velocity scale
∆T γ 0 h
.
(3.5)
µ1 + µ2 w
The force due to the tangential viscous shear is then obtained by multiplying σrθ ∼ µ2 U/h
by sin θ ' w/R and integrating on the portion w × h of the interface,
U∼
Ft ∼ µ2
µ2
hw
Uw
wh =
∆T γ 0
.
hR
µ1 + µ2
R
75
(3.6)
0
h=0.05
h/R=0.05
F.R/h
h=0.1
h/R=0.1
(c)
h=0.2
h/R=0.2
h/R=0.5
h=0.5
!1
normal shear
stress
!2
pressure
tangential
shear stress
total force
!3
0
0.2
0.4
0.6
0.8
1
w/R
Figure 3.8: Calculated force (numerical and analytical values) as a function of the hot
region size. Note that the force is negative (pushing the drop left) and increases with
decreasing drop size.
The force due to the normal viscous shear can be shown to scale like Fn ∼ Rh Ft and is
therefore negligible. The pressure force, on the other hand, derives from a balance between
the pressure gradient and the radial second derivative of velocity. In the present circular
geometry, similar scaling arguments yield a law for the contribution of the pressure force
Fp , which follows the same scaling as Ft , resulting in the same scaling law for the total
force F . A rigorous derivation [57] yields the final form of the force including the prefactor:
√
F = −2 π
µ2
hw
∆T γ 0
.
(3.7)
µ1 + µ2
R
This expression is represented (once non-dimensionalized) by the straight line on Fig. 3.8
and agrees very well with the numerically computed values.
The theoretical treatment brings out two length scales, h/R and w/R. While h and
w can be thought of as determining the typical scales for velocity variations in the radial
and azimuthal directions, R enters the force scaling as a local radius of curvature rather
than the actual size of the drop. It is therefore not surprising that the blocking force
should increase as R decreases. On the other hand, the drag force due to the external
flow scales as R2 [90], implying that the laser power necessary to counterbalance the drag
quickly decreases with the drop size.
3.3
Detailed measurements
Although the analytical study above provides support for the thermocapillary origin of
the force, many questions remain open regarding the applicability of the technique and on
the detailed physical mechanisms at play. For this, we undertook a series of measurements
in order to determine three parameters: (i) the dynamics and statics of the thermal field,
(ii) the dynamics of the flow field setup, and (iii) the distribution of surfactant in the oil
phase.
76
glass slides
aqueous solution
35 µm gap
microscope objective
dichroic mirror
laser
dichroic mirror
fluorescent lamp
filter
to fast camera
mirror
filter
Figure 3.9: Temperature measurement setup.
These studies were undertaken at LadHyX using an infrared laser (wavelength λ =
1480 nm) which directly heats the water molecules, therefore not requiring the addition
of a dye to provide the heating. This allowed us to use fluorescence to obtain physical
measurements. This work is currently being written up for publication [38, 121].
3.3.1
Temperature field measurements
The temperature fields were measured in a quiescent water layer, sandwiched between
two glass plates, as shown in Fig. 3.9. For these measurements, we took advantage of
the sensitivity of the quantum yield of Rhodamine B to temperature, which decreases
with increasing temperature of the solution [99]. The calibration of the thermal response
of the Rhodamine fluorescence was first made by imposing a known temperature with a
water bath, after which we could map directly the fluorescent intensity to temperature.
For the temperature profile measurements, the aqueous solution of Rhodamine was
heated with the laser and images were taken at 500 frames per second, in order to observe
the temperature increase at the laser position. Care had to be taken, however, to account
for the depletion of the dye molecules due to thermophoresis (Soret effect) [48]. This was
achieved by normalizing the fluorescence of Rhodamine B with images of Rhodamine 101,
a similar molecule which is not sensitive to temperature.
The temperature profiles, as a function of distance r from the laser and time t, were
well fitted with a Lorentzian curve
T (r, t) = T0 +
∆T (t)
,
1 + (r/σ(t))2
(3.8)
where σ(t) gives the width of the profile at half height and ∆T (t) is the maximum
temperature rise as shown on Fig. 3.10, which also shows the evolution of ∆T and σ as
a function of time. We observe a rapid response of the maximum temperature, which
reaches its steady-state value after less than 10 ms, due to the small volume being heated.
The final Lorentzian profile is set up over a longer time scale, requiring a few tens of ms.
The steady state temperature profile was also measured and provided the value of
the maximum temperature as a function of distance from the laser beam. The values of
σ ranged between 10 and 30 µm for the laser powers used in our experiments.
77
80
2 ms
4 ms
16 ms
lorentzian fit
22
(a)
(b)
20
40
18
40
!T (C)
50
30
! (µm)
60
!T (C)
temperature (C)
70
50
20
16
14
20
12
40
10
0
0
30
0
100
200
radius (µm)
300
400
0
0
0.05
20
time (ms)
0.1
time (s)
0.15
10
40
0.2
8
0
0.2
0.4
time (s)
0.6
0.8
1
Figure 3.10: Profile of the temperature rise for the initial few milliseconds of heating
(left). Transient increase in the maximum temperature (middle) and the width of the
Lorentzian curve (right) as a function of time.
3.3.2
Time scale for activation
A fundamental limit on the applicability of the technique depends on the time required
for the activation of the thermocapillary process. One measure of this time scale may be
to ask how fast the drops can flow past the laser and still be stopped. This was tested in
the channel of Fig. 3.11, where drops are formed at the first T junction and transported
into the “test section”, where they are stopped by the laser, while the oil is allowed to
continue flowing through the bypass. A second oil inlet is used to vary the velocity of the
drops without affecting the formation frequency and size.
Qwater
(2)
Q oil
laser
Bypass
Qoil(1)
500 µm
Test section
Figure 3.11: Channel for measuring force and roll formation time scale.
The experiment consists of blocking the drop advance in the test section for different
total flowrates, Q = Q1oil + Q2oil + Qwater . The drops were seeded with fluorescent particles and velocity fields inside the drops were measured using Particle Image Velocimetry
(PIV), as shown in Fig. 3.12 where the steady state velocity field (Fig. 3.12(b)) is compared with a streak photo showing the recirculation rolls (Fig. 3.12(a)). We see that the
PIV captures well the flow field far from the laser position but gives poor results near the
hot spot, partly because of the very high velocities and in part due to the decline if the
fluorescence intensity.
In order to extract a time scale for the setup of the rolls, the flux ϕ(t) is measured
across the line `, drawn on Fig 3.12(b). By continuity, we know that the flux passing
through ` must return through the the region d near the interface (Fig. a). Noting
furthermore that the geometry of the rolls does not seem to vary, the measurement of
ϕ(t) yields a time scale for the setup of the thermocapillary flow near the interface.
78
(b)
(a)
d
l
y
d
x
100 µm
Figure 3.12: Sample PIV field and measurement region.
15
distance !x (µm)
formation time of rolls !R (ms)
20
15
10
5
2
3
4
5
6
total flowrate Q (nL/s)
7
10
5
0
0
8
2.1 nL/s
4.6 nL/s
5.4 nL/s
7.9 nL/s
0.1
0.2
time (s)
0.3
0.4
Figure 3.13: Time scale for setup of rolls (left) and mean position of the drop from
integrated PIV field (right).
This time scale, τR , is plotted on Fig. 3.13(left) for a fixed laser power (P = 76 mW)
and different flowrates Q. In this figure, we observe that the characteristic time decreases
as the drop velocity increases, which suggests that the time to form the rolls is dictated
by the time required to enter the Gaussian laser spot. Furthermore, the fastest drop that
can be stopped displays a flow setup time of 6 ms, which is close to the time required to
reach the maximum temperature from the heating experiments discussed above. These
results suggest that the limit on the speed of actuation of a drop is given by the time
required to heat the liquid enough to create the Marangoni rolls.
The PIV data can also be used to obtain the position of the drop’s center of mass,
as shown in Fig. 3.13(right) for a few representative drops, by integrating the velocity
data in time. The position thus measured is precise since it relies on the average of many
velocity vectors. We see that drops that approach slowly are slowed down continuously
to a stop (circles), while drops advancing more quickly go through an overshoot before
retracting back to their final position. Finally, the ultimate drop that is held remains
stationary for only about 100 ms, but never relaxes back to the equilibrium position.
Detailed analysis of these results suggests that, in addition to the heating and the
creation of a Marangoni flow, the drop must also adapt its shape to the flow conditions,
which involves a second, slower, time scale. This can be used to explain why the fastest
drop is stopped for a short period, but doesn’t remain blocked: If the front interface has
passed most of the laser spot, the shape adaptation will lead to the laser moving further
into the drop and therefore reducing the temperature gradient along the interface, thus
de-blocking the drop.
79
t=0
(a)
flow
Intensity (a.u.)
150
laser position
100
50
(c)
0
!50
0
50
y [µm]
t=0.3 s
Intensity (a.u.)
150
(b)
100
50
(d)
0
!50
0
50
y [µm]
Figure 3.14: Fluorescence images showing the surfactant concentration as the drop begins
to be heated. Parts (a) and (b) show the drop at the moment of heating and 300 ms
later. Parts (c) and (d) show fluorescence intensity cuts at the positions marked by the
solid and dashed lines on the images.
3.3.3
Surfactant transport
Finally, the third limitation on the technique is to understand the dependence on the particular fluids and surfactants. For this, we had to understand the source of the anomalous
Marangoni flow and the relation between the heating and the surfactant transport. Measurements made using the pendant drop technique confirmed the classical linear decrease
of the interfacial tension with temperature for our fluid solutions. This motivated us to
look at transient effects.
In the same test section of Fig. 3.11, we performed an experiment where the surfactant micelles were marked with Rhodamine 101, which acts as a co-surfactant to the
Span 80. As discussed in Section 3.3.1, Rhodamine 101 is not sensitive to temperature
changes. Furthermore, it was observed that the dye is not soluble in pure hexadecane but
that it was strongly solubilized when the hexadecane contained a large concentration of
surfactant. This “micellar solubility” implies that the surfactant molecules could be used
as tracers of the micelles. Furthermore, it was observed that the partition between the oil
and the water phases was very strongly biased towards the oil phase, which means that
the drops appear as dark regions in a fluorescent oil phase.
In Fig. 3.14, two snapshots are shown at the moment at which the drop reaches
the laser and 300 ms later. The drop, which travels from left to right, displays an initial
decrease in fluorescence very close to the laser, as shown by the solid line in part (c).
We interpret this decrease in fluorescence as a decrease in the surfactant concentration
at this location, which is coherent with the increase in surface tension. At later times,
the fluorescence field becomes strongly inhomogeneous, indicating strong redistributions
of the surfactant molecules. In particular, the rolls transport a thin highly concentrated
jet away from the drop interface.
These results do not to provide a complete scenario of surfactant redistribution,
80
which would be highly nonlinear and would involve several coupled phenomena. Instead,
they point to significant redistributions in surfactant concentration, which confirms that
the dynamics of the redistribution can play a major role in the surface tension variations.
3.4
What can you do with it?
Going back to the applications of the technique in real microfluidics situations, we now focus on the droplet operations that can be achieved by locally heating a water-oil interface
with a focused laser. Recall that Joanicot and Ajdari had identified certain key operations that had to be controlled in order for droplets to be viable as microfluidic reactors:
fabrication, sorting, storage, fusion, breakup, and trafficking. Above, we have seen that
fabrication of drops could be controlled, namely by controlling the frequency and size of
drops dynamically. The rest of this section demonstrates the implementation of the other
operations, to which we may add synchronization, an important operation when drops
must interact. In the next section, we also introduce more complex operations, such as
order switching, by using more complex optical techniques.
3.4.1
Drop fusion
The simplest operation to understand is the one that leads to the fusion of two droplets,
shown in Fig. 3.16. Drops in microchannels are stabilized by the presence of surfactant
through several mechanisms. Particularly, the presence of a large concentration of surfactants introduces an increased resistance to flow between the two interfaces, which thus
prevent the trapped oil film from draining. Other interactions between the surfactants
on the two interfaces can also retard the merging, for example through electrostatic or
steric repulsion.
The localized heating can play two major roles in inducing the fusion of two drops.
The first role is to deplete the surfactant molecules from the surface, as we have seen
above, while the second role is to produce recirculation rolls which can actively drain the
lubrication film between the two drops, as shown schematically in the sketch of Fig. 3.15.
Here, we show the arrows pointing away from the laser spot; although the flow in the plane
of the channel is pointing towards the hot region, it will be associated with a secondary
recirculation flow, normal to the plane, which drains the fluid near the interface away
from the hot spot.
Oil
Water
Water
Laser
Figure 3.15: Sketch of the fusion mechanism: The laser drains the lubrication oil film.
We observe that the heating may be used to induce droplet merging, either in the
case of isolated drops (left sequence in Fig. 3.16) or in the case of a compact flow of
droplets, as shown in the right sequence of the same figure. In the case of isolated drops,
81
t=0
Flow
t=0.28 s
t=0.44 s
Figure 3.16: Fusion in the case of isolated drops (left) and a train of drops (right). Note
that the localized heating produces localized fusion events and this compares favorably
with the results of Priest et al. [96]
the laser is first used to block the advance of the initial drop until the second one collides
with it, at which point the two drops advance together, merging when the interfaces pass
the laser spot. In the case of the compact flow of drops, the drops do not need to be
synchronized and a lower laser power can be used to induce the merging event.
3.4.2
Synchronization
Furthermore, the synchronization of drops in order to combine their contents is a major challenge for lab-on-a-chip operations. Passive synchronization and alternation of
drops from two sources was demonstrated by finely tuning the different water and oil
flowrates [4]. This approach, however, is only useful in the simplest cases where only
two droplet streams are involved and the downstream conditions are constant. A more
robust approach would be for a downstream drop to delay its formation and wait for the
upstream drop to catch up with it, at which point the two merge together. This corresponds, in our terms, to a combination of a valve and a fusion mechanism; once the two
building blocks exist, combining them becomes a simple matter as shown in Fig. 3.17.
Here, drops are formed at successive T-junctions and flow down the same exit channel. In the absence of the laser forcing (Fig. 3.17-left), the drop formation is not synchronized and neither do they merge if they do come into contact. The situation is different in
the presence of the laser, which holds the downstream interface in place (Fig. 3.17a) until
the upstream drop is formed and collides with it (Fig. 3.17b). Since the upstream drop
completely blocks the channel, the hydrodynamic drag on the two-drop system becomes
too large and the two drops start to flow again (Fig. 3.17c), merging together when their
touching interfaces pass the laser (Fig. 3.17d).
The valve and fusion actions here are performed with only one laser spot, showing
how the different building blocks may be superposed by combining the laser action with a
geometric constraint. This is done with no overhead in power or complexity with respect
to a single operation, demonstrating how the technique may be scaled to a complex
lab-on-a-chip involving many operations.
82
Laser off
flow
Flow
Laser on
(a)
(b)
(c)
(d)
(e)
Figure 3.17: A forming drop is blocked by the laser-valve (a) until a second drop, formed
upstream, collides with it (b). The collision liberates the front drop (c) and the two merge
when their interfaces pass the laser(d). Operating conditions are Qwater = 0.1 µL/min,
Qoil = 1 µL/min, P = 67 mW and ω0 = 5.2 µm. The laser position is represented by
the white circle.
3.4.3
Routing
The remaining steps after the formation and merging of drops are their transport and
division, which involve control over the route they follow at bifurcating channels. Two operations are demonstrated below: sampling a drop, i.e. dividing it into unequal daughter
droplets of calibrated size, and sorting. A sampler which uses a combination of channel
geometry and laser forcing is shown in Fig. 3.18. We see in it drops that are longer than
the channel width and that arrive at a symmetric bifurcation, carried by the oil phase. At
the bifurcation, the drops divide into two parts whose lengths in the daughter channels
we label L1 and L2 . We are interested in the ratio λ ≡ h(L1 − L2 )/(L1 + L2 )i which yields
λ = 0 for symmetric drops and λ = 1 for complete sorting. The brackets h·i here denote
an average over several drops.
In the absence of the laser (Fig. 3.18a), we measure λ = 0.022 ± 0.01 for our microchannel, corresponding to a slight asymmetry in the microfabrication. When the laser
is applied in front of one of the two exits, the water-oil interface is asymmetrically blocked
at the laser position for a time τb , while the other side continues to flow (Fig. 3.18b). After
τb , both sides of the drop continue forward into their respective channels, but the retardation of the right hand droplet tip produces an asymmetry in the breaking, measured
by an increase in λ. Since the blocking time τb increases with the laser power, so does
the asymmetry in the division, as shown in Fig. 3.18c.
Above a critical power (approximately 100 mW for the present configuration), the
drop does not divide but is always diverted into the opposite branch, as shown in Fig. 3.19.
This sorting operation may be understood by considering the length of the droplet upstream of the laser: If the upstream length decreases below a critical size of approximately
the channel width (corrected by the displacement of the laser with respect to the channel
center), the drop takes a circular shape and loses contact with the right hand wall. In
83
(a)
Laser off
1
L1
(c)
L2
(b)
Laser on
!="(L1!L2)/(L1+L2)#
Flow
0.8
0.6
0.4
0.2
0
60
80
100
120
Power (mW)
Figure 3.18: Dividing drops asymmetrically: (a) Without laser forcing, a drop at a
bifurcation divides into approximately equal daughter droplets. (b) By controlling the
laser power (here P = 93 mW), we control the pinning time of one side of the interface and
thus the asymmetry of the division. Main channel width is 200 µm and the time between
images is 0.2 s. Operating conditions are Qwater = 0.02 µL/min, Qoil = 0.2 µL/min,
and ω0 = 5.2 µm. (c) Daughter droplet size dependence on laser power (λ = 0 is for
symmetric drops and λ = 1 is for the sorter). The dashed line corresponds to the mean
value of λ in the absence of the laser.
this case, a tunnel opens for the oil to flow between the drop and the wall and the drop
does not divide anymore but is pushed into the left hand channel, as shown in Fig. 3.19.
Finally, a more useful sorting operation can be achieved as shown in Fig. 3.20. Here,
the channel outputs are designed asymmetrically, such that drops all passively flow to
the top exit channel in the absence of active manipulation. When the laser is applied,
the drop can be pushed downward such that its center of mass is now located in the
streamlines flowing into the other exit channel. In this way, drops can be selected based
on their content, size, or their number.
(a) Laser off
(b) Laser on
Flow
Figure 3.19: A drop without laser heating divides into equal parts. If the laser heating is
sufficiently strong, the drop can be completely redirected into one of the two channels.
84
(b) Laser on
(a) Laser off
Oil flow
Obstacle
250 µm
Figure 3.20: Drops naturally flow into the upper channel due to a geometric asymmetry.
If the laser is applied, the drops can be sent to the lower channel.
3.5
More advanced manipulation
The possibilities of droplet manipulation can be extended by combining microfluidics with
optical holographic techniques, thus taking advantage of the contactless nature of optical
manipulation. Indeed, we show below that the use of different laser patterns allows the
implementation of complex operations which are not possible with any other methods.
We begin by exploring the effect of the shape of the laser focus on the blockage of droplets,
while further we demonstrate implementations which show conceptually new operations
on drops in microchannels.
Holographic beam shaping was employed to generate the desired patterns of light [28].
We first investigated the minimum optical power, Pmin , required to block the advance of a
drop, for three different shapes: a Gaussian spot with a 1 µm waist, a straight line aligned
with the flow direction, and a straight line perpendicular to the flow direction. Both lines
were 2 µm in width and 200 µm in length. In these experiments, the microchannels were
similar to those shown in Fig. 3.11 with three inlets, two for oil and one for water. Droplet
(1)
size was determined by the ratio between the flow rates of the first oil inlet Qoil and the
(2)
water inlet Qwater , which were both kept constant. The second oil flow rate Qoil was used
(2)
(1)
to tune the total flow rate Qtot = Qoil + Qoil + Qwater , keeping in this way the size of the
droplets constant while their velocity varied with Qtot .
Gaussian
parallel
perpendicular
1
(c)
optical intensity (MW/cm2)
optical power (mW)
500
Gaussian
parallel
perpendicular
400
300
200
100
0
(d)
0.8
0.6
0.4
0.2
0
2
4
6
8
10
total flow rate (nL/s)
12
2
4
6
8
10
total flow rate (nL/s)
12
Figure 3.21: Influence of the shape of the laser action.
85
(a)
(b)
(c)
100 µm
Figure 3.22: Routing drops through three separate exit channels. By changing the holographic pattern, the drops can be directed into the left, right, or front channel.
The first observation, as the drops reach the laser beam, is that the water-oil interface adapts to the laser forcing, as seen in Fig. 3.21(a) and (b). When the line is parallel
to the direction of flow, the front interface is flattened and the drop stops after advancing
through a significant portion of the line. In the case of a line perpendicular to the flow
direction, the surface of the drop is even flatter than in the previous case, taking on the
shape of the line.
The minimum laser power for each of the laser distributions is plotted as a function
of the total flow rate in Fig. 3.21(c). It scales approximately linearly with the total flow
rate but the slopes of the curves and the values of Pmin differ for the three cases. The use
of a line perpendicular to the flow allows the blocking of drops at higher flow rates, up
to more than 10 nL/s. Conversely, even though a lower laser power is necessary to hold
the droplets in the case of a Gaussian spot, it was not possible to hold droplets for flow
rates higher than about 5 nL/s. This was also the case for the line parallel to the flow.
Moreover, if the pattern intensity is studied instead of the total laser power, the
minimum intensity Imin necessary to block a drop is found to be several times higher for a
Gaussian spot than for a line distribution (Fig. 3.21(d)). The perpendicular line is found
to block the drop for the lowest value of Imin . Note that in the case of line patterns, the
image is only fully formed in the focal plane of the microscope objective, i.e. measuring
a few microns in depth. For the Gaussian spot however, the pattern propagates through
the whole sample.
We now consider the applications of such holographic beam shaping and how single
spot applications can be extended. The first application is droplet routing, or actively
sending droplets into different directions at a trifurcation. Making use of the ability to
both dynamically switch the optical patterns projected into the microfluidic channel and
the ability to create extended patterns (in this case four spots) we can deflect droplets
through large angles and send them into preferred channels. This is shown using a four
way cross channel in Figs. 3.19(a)-(c) which shows the droplets being moved to the left,
straight, or to the right, respectively. The switching time of the droplets into a given
channel is limited only by the update speed of the Spatial Light Modulator (SLM), which
ranges from 30 to 60 Hz. It would be straightforward to extend this technique to active
sorting, by including some video processing and combining it with the hologram switching
(the holograms are pre-calculated and are merely changed based on which direction the
droplets need to move in). One could imagine the sorting being based on droplet size,
chemical composition, fluorescence measurements or simply the contents of a drop.
86
t=0s
t = 5.8 s
100 µm
t = 0.4 s
t = 8.1 s
t = 3.3 s
t = 8.7 s
Figure 3.23: The ability to change the hologram and to hold drops for long times allows
the switching of droplet order. Note that the lower drop can be held for very long times.
t=0s
t = 0.57 s
Flow
100 µm
t = 0.3 s
t = 1.43 s
Figure 3.24: Drops arrive from the left and can be shuttled between three laser positions.
The second example uses two line patterns to store droplets at a given point in
the channel while rerouting other droplets to move past the stored droplet. The first
line upstream is set to move a droplet into one side of the larger channel. The droplet
is then stored by the downstream line further along the channel. The first line is then
changed so as to move subsequent droplets in the flow past the first droplet, as shown
in Fig 3.23. Thus we can store and could interrogate the first droplet without the need
to stop the flow, which is important in order to obtain longer interrogation times. Note
that the ability to focus the laser to a small area on the drop allows real droplet-level
manipulation, contrary to electrical fields which produce a uniform forcing on a region of
the microchannel [4]. This is what allows the drop order to be inverted in this case.
The third example, shown in Fig. 3.24, is an extension of the second. Here we are
able to trap several droplets at once, first one, then two and finally three using three
lines of light. Again this is in the presence of droplets flowing through the channel. We
are then able to shuttle the droplets through the pattern, by turning the whole pattern
on and off, so the first droplet is lost and the second droplet takes its place and so on.
This allows large scale storage and controlled movement of many droplets simultaneously
which may be useful for offline analysis of many droplets, droplet re-ordering or droplet
“memory” applications.
87
PHYSICAL REVIEW E 75, 046302 共2007兲
Thermocapillary valve for droplet production and sorting
Charles N. Baroud,1,* Jean-Pierre Delville,2,† François Gallaire,3 and Régis Wunenburger2
1
LadHyX and Department of Mechanics, Ecole Polytechnique, F-91128 Palaiseau cedex, France
CPMOH, UMR CNRS 5798, Université Bordeaux 1, 351 Cours de la Libération, F-33405 Talence cedex, France
3
Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, 06108 Nice cedex, France
共Received 27 July 2006; published 5 April 2007兲
2
Droplets are natural candidates for use as microfluidic reactors, if active control of their formation and
transport can be achieved. We show here that localized heating from a laser can block the motion of a water-oil
interface, acting as a microfluidic valve for two-phase flows. A theoretical model is developed to explain the
forces acting on a drop due to thermocapillary flow, predicting a scaling law that favors miniaturization.
Finally, we show how the laser forcing can be applied to sorting drops, thus demonstrating how it may be
integrated in complex droplet microfluidic systems.
DOI: 10.1103/PhysRevE.75.046302
PACS number共s兲: 47.61.⫺k, 47.55.dm
Microfluidic droplets have been proposed as microreactors with the aim to provide high performance tools for biochemistry. Individual drops may be viewed as containing one
digital bit of information and the manipulation of a large
number of slightly differing drops would allow the testing of
a large library of genes rapidly and with a small total quantity of material 关1兴. In microchannels, drops are produced
and transported using a carrier fluid 关2兴 and typical channel
sizes allow the manipulation of volumes in the picoliter
range. Surfactant in the carrier fluid prevents crosscontamination of the drops through wall contact or fusion
关3,4兴. However, while the geometry of the microchannel may
be used to determine the evolution of drops and their contents 关3–5兴, the implementation of real lab-on-a-chip devices
hinges on the active control of drop formation and its evolution, which remains elusive.
In this paper, we remedy the situation by demonstrating
experimentally how a focused laser can provide precise control over droplets through the generation of a thermocapillary
flow. In doing so, we develop the first theoretical model of a
droplet subjected to localized heating, yielding a general understanding of the forces acting on the drop and a scaling law
which favors miniaturization. A carrier fluid is still used for
the formation and transport of drops but the effects of geometry are augmented with a local thermal gradient produced
by the laser beam, focused through a microscope objective
inside the microchannel.
Indeed, moving drops with heat has been a preoccupation
of fluid mechanicians since the initial work of Young et al.
关6兴. Although originally motivated by microgravity conditions where surface effects are dominant 关6,7兴, microfluidics
has opened up a new area where bulk phenomena are negligible compared to surface effects. Recently, thermal manipulation of drops or thin films resting on a solid substrate has
received the attention of the microfluidics community either
through the embedding of electrodes in the solid 关8,9兴 or
through optical techniques 关10–12兴. However, the physical
mechanisms in the transmission of forces when the liquid
*Electronic address: [email protected]
†
Electronic address: [email protected]
1539-3755/2007/75共4兲/046302共5兲
touches a solid wall are fundamentally different from the
case of drops suspended in a carrier fluid, away from the
boundaries 关13兴. The latter case has received little attention
despite the advantages that microchannels offer over open
geometries.
Our experimental setup consists of a microchannel fabricated using soft lithography techniques 关14兴. Water and oil
共hexadecane +2 % w / w Span 80, a surfactant兲 are pumped
into the channel at constant flow rates, Qwater and Qoil,
using glass syringes and syringe pumps. Channel widths are
in the range 100– 500 ␮m and the height h is in the range
25– 50 ␮m. Local heating is produced by a continuous
argon-ion laser 共wavelength in vacuum ␭Ar+ = 514 nm兲, in the
TEM00 mode, focused inside the channel through a ⫻5 or
⫻10 microscope objective to a beam waist ␻0 = 5.2 or
2.6 ␮m, respectively. The optical approach can be reconfigured in real time and it allows the manipulation inside small
microchannels with no special microfabrication. The absorption of the laser radiation by the aqueous phase is induced by
adding 0.1% w / w of fluorescein in the water.
A surprising effect is observed when the water-oil interface reaches the laser spot: In the cross-shaped microchannel
of Fig. 1, we produce water drops in oil through the hydrodynamic focusing technique in which two oil flows pinch off
water droplets at the intersection of the channels. In the absence of the laser, drops of water are produced in a steady
fashion and are transported along with the oil down the drain
channel, as shown in Figs. 1共a兲–1共c兲. When the laser is illuminated, however, the oil-water interface is blocked in place
as soon as it crosses the beam. While the typical drop pinching time is ␶d ⬃ 100 ms in the absence of the laser, we find
that we can block the interface for a time ␶b, which may be
of several seconds, as shown in Figs. 1共d兲–1共f兲共see Ref. 关15兴
for video sequences兲. During the time ␶b, the drop shedding
is completely inhibited and the volume in the water tip increases until the viscous stresses finally break it off. The drop
thus produced is larger, since it has been “inflated” by the
water flow.
In the microchannel shown in Fig. 2, we measured the
variation of the blocking time ␶b with respect to laser power
and forcing position. We observe that ␶b increases approximately linearly with the power, above an initial threshold,
046302-1
©2007 The American Physical Society
Laser off
(a)
(b)
(c)
Laser on
BAROUD et al.
(d)
(e)
(f)
Laser
FIG. 1. Microfluidic valve: In a cross-shaped microchannel, the oil flows from the lateral channels and the water enters through the
central channel. 共a兲–共c兲 In the absence of laser forcing, drops are shed with a typical break-off time 关共b兲 and 共c兲兴 of 0.1 s. 共d兲–共f兲 When the
laser is applied, the interface is blocked for several seconds, producing a larger drop. In image 共e兲, the evolution of the neck shape is shown
through a superposition of 100 images 共2 s兲. Exit channel width is 200 ␮m. Operating conditions are: Qwater = 0.08 ␮L / min,
Qoil = 0.90 ␮L / min, beam power P = 80 mW, and beam waist ␻0 = 5.2 ␮m.
showing a weak position dependence of the laser spot. Furthermore, the inset of Fig. 2 shows that the droplet length L
varies linearly with ␶b. As expected from mass conservation
at constant water flow rate, L = L0 + ␶bQwater / S, L0 being the
droplet length without laser and S ⯝ 共125⫻ 30兲 ␮m2 the
channel cross section. The best linear fit to the data gives an
effective water flow rate Qwater = 0.028 ␮L / min, close to the
nominal value 0.03 ␮L / min, showing that the water flow
rate remains controlled even in presence of the laser forcing.
FIG. 2. 共Color online兲 Dependence of the blocking time ␶b
on laser power and position 共indicated in the picture兲 for
Qwater = 0.03 ␮L / min and Qoil = 0.1 ␮L / min, ␻0 = 2.6 ␮m. Inset:
Rescaled droplet length L / L0 vs the blocking time 共laser position •兲,
where L0 is the droplet length without the laser. The dashed line is
a linear fit, ignoring the outlier at ␶b = 0.75 s.
Thus, the optical forcing provides a tunable valve, which
provides control over droplet timing and size. Similar blocking is observed in a T geometry or if the flows are driven at
constant pressure. However, the blocking is only obtained
when the light is absorbed, here through the use of a dye.
We visualize the convection rolls produced by the heating
by placing tracer particles in both fluids, as shown in Fig.
3共a兲, for a drop that is blocked in a straight channel. For pure
liquids, the direction of Marangoni flow along the interface
is directed from the hot 共low surface tension兲 to the cold
共high surface tension兲 regions. However, the flows in our
experiments point towards the laser along the interface, indicating an increase of surface tension with temperature. This
is consistent with previous studies that have shown a linear
increase of surface tension with temperature in the presence
of surfactants 关16–18兴.
One important constraint for practical applications
is the amplitude of the temperature rise. Since the materials
used in this study have similar thermal properties 共thermal
thermal
conductivity
diffusivity
Dth ⬃ 10−7 m2 s−1,
⌳th ⬃ 0.5 W m−1 K−1兲, we estimate the maximum temperature in the flow by modeling the heating produced by a laser
absorbed in a single fluid phase 关19兴, assuming thermal diffusion as the only energy transport mechanism. Considering
the measured optical absorption of our water/dye solution
␣th = 117.9 m−1, and assuming that the temperature 100 ␮m
away is fixed by the flowing oil at room temperature, we find
⌬T ⯝ 12 K for the temperature rise at the laser focus for a
beam power P = 100 mW. The temperature gradient is steep
near the focus, with the temperature dropping to 5 K at
20 ␮m from the beam spot. However, note that given the
typical flow velocity 共U ⬃ 1 mm/ s兲 and the characteristic
length scale over which thermal diffusion occurs
共L = 100 ␮m兲, the thermal Péclet number Pe= UL / Dth is
comparable to unity. Thus, our calculation overestimates the
actual overheating.
046302-2
THERMOCAPILLARY VALVE FOR DROPLET PRODUCTION…
冉
1 ⳵2
1 ⳵ ⳵
r + 2 2
r ⳵r ⳵r r ⳵␪
冊冉
冊
1 ⳵2 12
1 ⳵ ⳵
r + 2 2 − 2 ␺ = 0, 共1兲
r ⳵r ⳵r r ⳵␪
h
where the depth-averaged velocities may be retrieved from
u␪ = −⳵␺ / ⳵r and ur = 1 / r共⳵␺ / ⳵␪兲. The kinematic boundary
conditions at the drop interface 共r = R兲 are zero normal velocity and the continuity of the tangential velocity. The normal dynamic boundary condition is not imposed since the
drop is assumed to remain circular, which is consistent with
our experimental observations, Fig. 3共a兲. Finally, the tangential dynamic boundary condition, which accounts for the optically induced Marangoni stress, is
冉冊
冉冊
⳵ u␪1
⳵ u␪2
␥⬘ dT
␮ 1r
− ␮ 2r
=−
,
⳵r r
⳵r r
r d␪
2
1.5
main flow
1
0.5
θ
r
x
0
y
−0.5
−1
(a)
−1.5
(b)
−2
−2
−1.5
−1
−0.5
0
0
0.5
h=0.05
h/R=0.05
h=0.1
h/R=0.1
1
1.5
2
(c)
h=0.2
h/R=0.2
h/R=0.5
h=0.5
−1
normal shear
stress
−2
pressure
共2兲
tangential
shear stress
total force
where ␮1,2 are the dynamic viscosities and u␪1,2 are the velocities in the drop and the carrier fluid, respectively.
␥⬘ = ⳵␥ / ⳵T is the surface tension to temperature gradient,
which is positive in our case.
For simplicity, we approximate the steady-state
temperature distribution using a Gaussian form T共x , y兲
= ⌬T exp兵−关共x − R兲2 + y 2兴 / w2其, where ⌬T is the maximum
temperature difference between the hot spot and the far field
and w corresponds to the size of the diffused hot spot, which
is significantly larger than ␻0 关19兴. The equations are nondimensionalized using ⌬T as temperature scale, R as length
R 共 ␮ 1+ ␮ 2兲
laser
F.R/h
The force generated by the convective flow on a droplet is
investigated through the depth-averaged Stokes equations,
since our channels have a large width/height aspect ratio
关20兴. The detailed modeling will be discussed in a subsequent publication; here we limit ourselves to the main features: a circular drop of radius R is considered in an infinite
domain and the flow due to the Marangoni stresses is evaluated. Assuming a parabolic profile in the small dimension 共h兲
and introducing a stream function for the mean velocities in
the plane of the channel, the depth-averaged equations, valid
in each fluid, are
scale, R␥⬘⌬T as force scale, and ␥⬘⌬T as time scale, the
remaining nondimensional groups being the aspect ratio h / R,
the nondimensional hot spot size w / R, and the viscosity ratio
¯ = ␮2 / 共␮1 + ␮2兲.
␮
A typical predicted flow field solving the above numerical
formulation is shown in Fig. 3共b兲, in which the four recirculation regions are clearly visible. The velocity gradients display a separation of scales in the normal and tangential directions, as observed from the distance between the
streamlines in the two directions. Indeed, it may be verified
that the velocities vary over a typical length scale h / R in the
normal direction, while the tangential length scale is given
by w / R.
Along with this flow field, we compute the pressure field,
⳵ū
¯ ⳵r̄r̄ 兲 and tangential
as well as the normal 共¯␴r̄r̄ = 2␮
关¯␴r̄␪ = ␮¯ 共 1r̄ ⳵⳵ū␪r̄ + ⳵⳵ūr̄␪ − ūr̄␪ 兲兴viscous shear stresses in the external
flow. Their projections on the x axis, shown in the inset of
Fig. 3共c兲, are then summed and integrated along ␪ to yield
the total dimensionless force 共F̄兲 on the drop. Note that the
global x component of the force is negative and therefore
−3
0
0.2
0.4
0.6
0.8
1
w/R
FIG. 3. 共Color online兲共a兲 Overlay of 100 images from a video
sequence showing the motion of seeding particles near the hot spot.
Note that the motion along the interface is directed towards the hot
spot. 共b兲 Stream function contours obtained from the depthaveraged model described in the text. Dashed and continuous contours indicate counterclockwise and clockwise flows, respectively.
共c兲 Rescaled nondimensional force F̄R / h plotted as a function of
¯ = 3 / 4. The straight line
w / R for various aspect ratios h / R for ␮
corresponds to the dimensional scaling derived in the text. The inset
shows the x component of the distribution along the azimuthal direction of the pressure, normal and tangential shear stresses, where
the solid circle is the reference zero. Their sum yields the total
force. Channel width in part 共a兲 is 140 ␮m. h / R = 0.2, w / R = 0.5 for
parts 共b兲 and 共c兲 inset.
opposes the transport of the drop by the external flow. The y
component vanishes by symmetry and the integral of the
wall friction may be shown to be zero since the drop is
stationary. Numerically computed values of F̄R / h are shown
by the isolated points in Fig. 3共c兲 as a function of w / R, for
different values of the aspect ratio h / R. The points all collapse on a single master curve, displaying a nondimensional
scaling law F̄ ⬀ wh / R2, for small w / R.
The dimensional form of the force can be obtained, for
small h / R and w / R, by considering the three contributions
separately and noting that the velocity scale in this problem
is imposed by the Marangoni stress. Using the separation of
scales along the azimuthal and radial directions, Eq. 共2兲 beR
comes 共␮1 + ␮2兲 Uh ⬃ ⌬TR␥⬘ w , where the ‘‘⬃’’ is understood as
046302-3
BAROUD et al.
F = − 2 冑␲
FIG. 4. Sorting drops: 共a兲 Without laser forcing, a drop at a
bifurcation divides into approximately equal daughter droplets. 共b兲
When the laser forcing is applied, the drop advance in the righthand channel is blocked so the whole drop is diverted into the left
channel. Main channel width is 200 ␮m and the operating
conditions are Qwater = 0.02 ␮L / min, Qoil = 0.2 ␮L / min, and
␻0 = 5.2 ␮m.
an order-of-magnitude scaling. This yields the characteristic
tangential velocity scale
U⬃
⌬T␥⬘ h
.
␮1 + ␮2 w
共3兲
The force due to the tangential viscous shear is then obtained
by multiplying ␴r␪ ⬃ ␮2U / h by sin ␪ ⯝ w / R and integrating
on the portion wh of the interface,
Ft ⬃ ␮2
Uw
␮2
hw
wh =
⌬T␥⬘ .
hR
␮1 + ␮2
R
共4兲
The force due to the normal viscous shear can be shown to
scale like Fn ⬃ Rh Ft and is therefore negligible. The pressure
force, on the other hand, derives from a balance between the
pressure gradient and the radial second derivative of velocity.
In the present circular geometry, similar scaling arguments
yield a law for the contribution of the pressure force F p,
which follows the same scaling as Ft, resulting in the same
scaling law for the total force F. A rigorous derivation 共to be
published elsewhere兲 yields the final form of the force including the prefactor
关1兴 O. Miller, K. Bernath, J. Agresti, G. Amitai, B. Kelly, E. Mastrobattista, V. Taly, S. Magdassi, D. Tawfik, and A. Griffiths,
Nat. Methods 3, 561 共2006兲.
关2兴 T. Thorsen, R. W. Roberts, F. H. Arnold, and S. R. Quake,
␮2
hw
⌬T␥⬘ .
␮1 + ␮2
R
共5兲
This expression is represented 共once nondimensionalized兲 by
the straight line on Fig. 3共c兲 and agrees very well with the
numerically computed values.
The physical value of the force for a typical experiment is
estimated by taking ␮1 = 10−3 N m−2 s 共water兲, ␮2 = 3␮1
共hexadecane兲, and extracting ␥⬘ ⬃ 1 mN m−1 K−1 from Ref.
关17兴. This yields a force on the order of 0.1 ␮N, which is of
the same order as the drag force on a drop in a large aspect
ratio channel 关21兴, thus confirming that thermocapillary forcing can indeed account for the blocking. Note that the force
we calculate is several orders of magnitude larger than those
generated from electric fields 关22兴 or optical tweezers 关23兴.
This blocking force may be applied at different locations
in a microchannel by displacing the laser spot. In particular,
we demonstrate the sorting of drops, a fundamental operation
in the implementation of lab-on-a-chip devices. Drops are
formed, as above, in a cross junction and arrive at a symmetric bifurcation, carried by the continuous phase. In the absence of laser forcing, the drops arriving at the bifurcation
divide into two equal parts 关5兴关Fig. 4共a兲兴. When the laser is
applied, the water-oil interface is asymmetrically blocked on
the right-hand side while the left-hand side continues to flow
关Fig. 4共b兲兴. Above a critical laser power 共approximately
100 mW for the present configuration兲, the drop is blocked
long enough that it is completely diverted through the lefthand channel 关15兴. Drops may therefore be sorted by accordingly selecting the laser position.
In summary, we have experimentally and theoretically
demonstrated the efficiency of laser-driven blocking of
water-in-oil drops. The theoretical treatment brings out two
length scales, h / R and w / R. While h and w can be thought of
as determining the typical scales for velocity variations in the
radial and azimuthal directions, R enters the force scaling as
a local radius of curvature rather than the actual size of the
drop. It is therefore not surprising that the blocking force
should increase as R decreases. On the other hand, the drag
force due to the external flow scales as R2 关21兴, implying that
the laser power necessary to counterbalance the drag quickly
decreases with the drop size. This, along with the rapidity of
viscous and thermal diffusion while thermal inertia is reduced, all lead to laws favorable to miniaturization. The generality of the process provides a practical new way for acting
on individual droplets, at any location, while working inside
the robust environment of the microchannel.
We acknowledge help from Julien Buchoux, David Dulin,
and Emilie Verneuil. This work was partially funded by the
CNRS PIR “Microfluidique et Microsystèmes Fluidiques,”
the Conseil Régional d’Aquitaine, and the convention
X-DGA.
Phys. Rev. Lett. 86, 4163 共2001兲.
关3兴 H. Song, J. Tice, and R. Ismagilov, Angew. Chem., Int. Ed.
42, 768 共2003兲.
关4兴 M. Joanicot and A. Ajdari, Science 309, 887 共2005兲.
046302-4
THERMOCAPILLARY VALVE FOR DROPLET PRODUCTION…
关5兴 D. R. Link, S. L. Anna, D. A. Weitz, and H. A. Stone, Phys.
Rev. Lett. 92, 054503 共2004兲.
关6兴 N. Young, J. Goldstein, and M. Block, J. Fluid Mech. 6, 350
共1959兲.
关7兴 R. Balasubramaniam and R. Subramanian, Phys. Fluids 12,
733 共2000兲.
关8兴 T. Sammarco and M. Burns, AIChE J. 45, 350 共1999兲.
关9兴 A. Darhuber, J. Valentino, J. Davis, S. Troian, and S. Wagner,
Appl. Phys. Lett. 82, 657 共2003兲.
关10兴 N. Garnier, R. O. Grigoriev, and M. F. Schatz, Phys. Rev. Lett.
91, 054501 共2003兲.
关11兴 J. Sur, T. P. Witelski, and R. P. Behringer, Phys. Rev. Lett. 93,
247803 共2004兲.
关12兴 K. Kotz, K. Noble, and G. Faris, Appl. Phys. Lett. 85, 2658
共2004兲.
关13兴 E. Lajeunesse and G. Homsy, Phys. Fluids 15, 308 共2003兲.
关14兴 D. Duffy, J. McDonald, O. Schueller, and G. Whitesides, Anal.
Chem. 70, 4974 共1998兲.
关15兴 See EPAPS Document No. E-PLEEE8-75-059703 for movie
关16兴
关17兴
关18兴
关19兴
关20兴
关21兴
关22兴
关23兴
046302-5
sequences from the experiments. For more information on
EPAPS, see http://www.aip.org/pubservs/epaps.html.
B. Berge, O. Konovalov, J. Lajzerowicz, A. Renault, J. P. Rieu,
M. Vallade, J. Als-Nielsen, G. Grübel, and J. F. Legrand, Phys.
Rev. Lett. 73, 1652 共1994兲.
E. Sloutskin, C. Bain, B. Ocko, and M. Deutsch, Faraday Discuss. 129, 1 共2005兲.
A reduction of the surfactant concentration reduced the efficiency of the blocking; a detailed exploration of surfactant
effects will be addressed in a future study.
J. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R.
Whinnery, J. Appl. Phys. 36, 3 共1965兲.
W. Boos and A. Thess, J. Fluid Mech. 352, 305 共1997兲.
A. Nadim, A. Borhan, and H. Haj-Hariri, J. Colloid Interface
Sci. 181, 159 共1996兲.
K. Ahn, C. Kerbage, T. Hynt, R. Westervelt, D. Link, and D.
Weitz, Appl. Phys. Lett. 88, 024104 共2006兲.
D. Grier, Nature 共London兲 424, 810 共2003兲.
PAPER
www.rsc.org/loc | Lab on a Chip
An optical toolbox for total control of droplet microfluidics{{
Charles N. Baroud,*a Matthieu Robert de Saint Vincentb and Jean-Pierre Delville*b
Received 15th February 2007, Accepted 17th May 2007
First published as an Advance Article on the web 8th June 2007
DOI: 10.1039/b702472j
The use of microfluidic drops as microreactors hinges on the active control of certain fundamental
operations such as droplet formation, transport, division and fusion. Recent work has
demonstrated that local heating from a focused laser can apply a thermocapillary force on a liquid
interface sufficient to block the advance of a droplet in a microchannel (C. N. Baroud, J.-P.
Delville, F. Gallaire and R. Wunenburger, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.,
2007, 75(4), 046302). Here, we demonstrate the generality of this optical approach by
implementing the operations mentioned above, without the need for any special microfabrication
or moving parts. We concentrate on the applications to droplet manipulation by implementing a
wide range of building blocks, such as a droplet valve, sorter, fuser, or divider. We also show how
the building blocks may be combined by implementing a valve and fuser using a single laser spot.
The underlying fundamentals, namely regarding the fluid mechanical, physico-chemical and
thermal aspects, will be discussed in future publications.
1. Introduction
Droplets are natural candidates for use as microreactors, since
they transport fluid with no dispersion and may be formed and
manipulated using microfluidic techniques.1–4 Indeed, a drop
may be formed with a known composition and volume5–7 and
transported by an inert fluid without loss of the solute species
and without cross-contamination.8 Furthermore, fusion of two
drops containing two reactive species leads to the onset, on
demand, of a reaction9 whose product may be sampled by
breaking the drop at a bifurcation.10 Finally, logical operations can be performed on drops by sorting them based on a
test of their contents, as they reach a bifurcation in the
microchannel.11,12 The above operations form the basis of a
droplet-based lab-on-a-chip which can be designed through an
intelligent combination of a few building blocks. Conversely, a
lack of active control over individual drops would severely
limit the usefulness of droplet microreactors.
However, acting on individual drops in microchannels
remains difficult. Recent publications have demonstrated, via
electrode micropatterning on the chip, the use of electric fields
to apply forces on droplets11 or to merge them.9,13 However,
the forces generated through dielectrophoresis were measured
to be in the range of a few nN and scale with the cube of the
drop radius, since the electrophoretic force is a body force.11
This is a highly unfavourable scaling which implies that the
force generated will quickly decrease as the drop size decreases.
In contrast, others have demonstrated the use of surface forces
a
LadHyX and Department of Mechanics, Ecole Polytechnique, 91128,
Palaiseau cedex, France. E-mail: [email protected]
b
Université Bordeaux 1, CPMOH (UMR CNRS 5798), 351 Cours de la
Libération, F-33405, Talence cedex, France.
E-mail: [email protected]
{ Electronic supplementary information (ESI) available: Supporting
videos 1 (droplet fusion), 2 (valving and fusion) and 3 (division and
sorting). See DOI: 10.1039/b702472j
{ The HTML version of this article has been enhanced with colour
images.
This journal is ß The Royal Society of Chemistry 2007
to manipulate drops on open substrates by modulating their
surface properties chemically, electrically or thermally (see e.g.
ref. 14 and references therein). These surface forces become
dominant over body forces at small scales, as the ratio of
surface to volume becomes large. It is natural therefore to look
for a surface mechanism for the manipulation of drops inside
the robust environment of a microchannel.
Along these lines, we have recently demonstrated that forces
near the mN range could be produced on a droplet by optically
heating a water–oil–surfactant interface with a laser wave.12
This force is generated through the thermocapillary (or
Marangoni) effect, by which the surface tension varies due to
a temperature variation; localized heating from a focused laser
therefore leads to a spatial imbalance of surface tension which,
in turn, induces a flow inside and around the drop. By
computing the shear and pressure fields associated with the
external flow, one may find that a net force is produced on the
drop.15 A theoretical analysis for localized heating shows a
scaling that is highly favourable to miniaturization, since the
total force is predicted to increase as the drop radius decreases.12
In our experiments, we observe that the surface tension rises
as the temperature is increased. This anomalous behaviour,
likely due to the presence of surfactant,16,17 yields a force that
pushes the drop away from the hot spot and acts to block it in
the presence of an external carrier flow. Since the time required
for the Marangoni flow to appear is short enough, a droplet
can be blocked during its formation, corresponding to a
contactless optical microfluidic valve, which can also be used
to control the size of the drops thus produced. Finally, drops
can also be sorted by simply illuminating one exit of a
bifurcating microchannel.12
Below, we show the generality of this optical approach and
how it may be used to provide a complete set of tools for the
manipulation of drops in microchannels. These tools allow
the control of drop formation and sorting, as previously
demonstrated, and also drop fusion and division. We also
Lab Chip, 2007, 7, 1029–1033 | 1029
demonstrate how the operations may be combined, while still
using a single laser spot, through a combination of channel
geometry and laser actuation. This opens the way for total
control of droplet microreactors without the need for specific
microfabrication.
2. Experimental
Our experimental setup consists of a microchannel fabricated in
Polydimethylsiloxane (PDMS), using standard soft lithography
techniques. Water and oil (hexadecane + 2% w/w Span 80, a
surfactant) are pumped into the channel at constant flowrates,
Qwater and Qoil, using glass syringes and syringe pumps; the
fluids may also be forced at constant pressure. Channel widths
are in the range 100–500 mm and the height is in the range 25–
50 mm. Local heating is produced by focusing a continuous
Argon-Ion laser (wavelength in vacuum lAr+ = 514 nm), in the
TEM00 mode through a microscope objective. The absorption
of the laser is mediated by the addition of a dye, such as
fluorescein (0.1% w/w), in the water phase. The resulting optical
absorption of the aqueous phase is about 1.18 cm21.
Different microscope objectives were used to focus the laser
inside the microchannel, ranging from a 62.5 to a 610
magnification, which correspond to beam waists (v0) in the
range of 10.3 to 2.6 mm. The Fresnel length, defined as LF =
npv02/l where n is the refractive index and l the wavelength in
vacuum, may be estimated at LF. 50 mm by using n = 1.33 and
v0 = 2.5 mm. Consequently, we can assume that the focused
beam is almost cylindrical over a distance of 100 mm (50 mm on
each side of the beam waist), which is twice the largest thickness
of our channels. This implies that the use of low magnification
objectives makes the behavior rather insensitive to the exact
focus plane, as opposed for example to laser tweezers.
3. Formation and fusion
3.1 Microfluidic valve
The valve mechanism for two-phase flows was recently achieved12
by illuminating the water–oil interface during the drop formation
at a cross-junction, with a laser power (P) on the order of a few
tens of mW, focused through a microscope objective. The local
heating thus produced was shown to completely block the
advance of the interface for a time tb which increased with
increasing laser power. This blocking also provided control over
the size of the drops thus produced, since they were inflated by the
syringe pumps operating at a constant flowrate.
This valve is generic and works equally well in a T geometry
where the oil and water arrive either from opposite channels
(Fig. 1a) or from perpendicular channels (e.g. Fig. 4). Similar
blocking is also observed if the flows are driven at constant
pressure or by mixing pressure and flowrate sources. For
instance, Fig. 1a shows the laser blocking the drop shedding at
different locations with the oil flow (bottom channel) driven
at constant flowrate and the water flow (top channel) driven at
constant pressure. In the absence of the laser, drops are formed
in a periodic fashion. In the presence of the laser, the water
interface is blocked at the laser focus, as shown in the Figure,
while the oil continues to flow. The variation of the blocking
time tb with the laser power and position is illustrated in
1030 | Lab Chip, 2007, 7, 1029–1033
Fig. 1 Microfluidic valve in a T geometry. (a) Superposition of
microscopic images of the laser blocking the interface at different
locations in the microchannel. All channels are 100 mm wide. (b)
Dependence of the blocking time tb on laser power and position
(indicated in (a)) for Qoil = 0.05 mL min21 and Pwater = 2.3 6 103 Pa,
v0 = 2.6 mm. The lines are linear fits to guide the eye.
Fig. 1b. While tb increases approximately linearly with the
power above an initial threshold, it also displays a dependence
on the laser position in the microchannel. The values of the
onset and the slope of tb depend on the details of the flow, but
the same general behaviour is observed independently of the
microchannel geometry, flowrates, or pumping method.
The dependence of the blocking time on beam waist was
explored in a cross-geometry by keeping constant the fluid flow
rates (Qwater = 0.12 mL min21 and Qoil = 0.3 mL min21) and the
laser position. The geometry that was used corresponds to a
cross-junction with oil channel widths 100 mm and water
channel width 200 mm. The laser was placed at a distance 200 mm
downstream of the oil channel centerline and the blocking time
(tb) was measured as a function of beam waist, which was varied
by changing the microscope objective. The measurements of tb
were normalized by the natural emission frequency of the drops
(F0) and were fitted by straight lines to determine the threshold
power (Pth) and slope (S = F0dtb/dP). While Pth was found to
remain constant at Pth . 40 mW, S increased with decreasing
beam waist as S = 4.5 6 1023, 8.2 6 1023 and 13.4 6 1023 W21
for v0 = 10.3, 5.2 and 2.6 mm, respectively.
3.2 Fusion of drops
Fusing droplets is the step that allows chemical reactions by
bringing together the reactants originally contained in separate
drops. However, simply putting drops in contact is typically
insufficient to induce merging, since a lubrication film between
them prevents the water contained in the two drops from
actually touching. Indeed, the presence of surfactant molecules
on an oil–water interface is known to stabilize drops against
merging.18,19 Localized heating close to the nearly touching
interfaces may be used to evacuate the surfactant molecules and
with them the oil film, as shown in Fig. 2. Here, the downstream
drop is held stationary by the laser heating until a second one
collides with it (Fig. 2a). At this point, the two drops advance
until the laser gets near the adjacent interfaces, and we observe
that the oil film is evacuated and the two drops rapidly merge.
Similar merging may be obtained in a long train of drops, as
shown in Fig. 3. Here, a train of water drops is carried by an
oil flow in a wide channel. Again, these drops are stable
against merging due to the presence of the surfactant and
spontaneous merging is never observed in our experiments
(Figs. 3a,b). However, weak heating at the interface from the
laser spot, although insufficient to block the drop advance,
rapidly induces merging when the laser spot approaches the
adjacent interfaces (Fig. 3c). Merging only occurs in the heated
region (Fig. 3d) while the other interfaces remain unaffected.
This shows that one may induce the merger of specific drops
even in a complex flow which contains many drops and
interfaces. A succession of such events is shown in the
supporting video 1 in the ESI{.
3.3 Combined operations: drop fusion at formation
The synchronization of drops in order to combine their
contents is a major challenge for lab-on-a-chip operations.
Alternating formation of drops from two sources was recently
demonstrated by finely tuning the different water and oil
flowrates.9 This approach, however, is only useful in the
simplest cases where only two droplet streams are involved and
Fig. 2 Time sequence showing droplet fusion through laser heating.
(a) The blocking of a first drop by the laser brings the drop that
follows in contact with it. (b) The two drops move forward together,
their coalescence occurring when the beam reaches the touching
interfaces, giving birth to a larger drop (c). Time between images is
40 ms and operating conditions are Qwater = 0.2 mL min21, Qoil =
0.9 mL min21, P = 67 mW and v0 = 2.6 mm.
Fig. 3 Localized fusion in a train of large drops. The drops, which
flow from left to right, merge as the interface crosses the laser. Time
between images is 30 ms and operating conditions are Qwater =
0.2 mL min21, Qoil = 0.3 mL min21, P = 67 mW and v0 = 5.2 mm.
the downstream conditions are constant. A more robust
approach would be for a downstream drop to delay its
formation and wait for the upstream drop to catch up with it,
at which point the two merge together. This corresponds, in
our terms, to a combination of a valve and a fusion
mechanism; once the two building blocks exist, combining
them becomes a simple matter as shown in Fig. 4.
Here, drops are formed at successive T-junctions and flow
down the same exit channel. In the absence of the laser forcing
Fig. 4 A forming drop is blocked by the laser-valve (a) until a second
drop, formed upstream, collides with it (b). The collision liberates the
front drop (c) and the two merge when their interface approach
the laser (d). Operating conditions are Qwater = 0.1 mL min21, Qoil =
1 mL min21, P = 67 mW and v0 = 5.2 mm. The laser position is
represented by the white circle.
Lab Chip, 2007, 7, 1029–1033 | 1031
(not shown), the drop formation is not synchronized and
neither do they merge if they do come into contact. The
situation is different in the presence of the laser, which holds
the downstream interface in place (Fig. 4a) until the upstream
drop is formed and collides with it (Fig. 4b). Since the
upstream drop completely blocks the channel, the hydrodynamic drag on the two-drop system becomes too large and the
two drops start to flow again (Fig. 4c), merging together when
their touching interfaces approach the laser (Fig. 4d). (See
supporting video 2 in the ESI{.)
The valve and fusion actions here are performed with only
one laser spot, showing how the different building blocks may
be superposed by combining the laser action with a geometric
constraint. This is done with no overhead in power or
complexity with respect to a single operation, demonstrating
how the technique may be scaled to a complex lab-on-a-chip
involving many operations.
4. Drop transport: division and sorting
The remaining steps after the formation and merging of drops
are their transport and division, which involve control over the
route they follow at bifurcating channels. Two operations are
demonstrated below: sampling a drop, i.e. dividing it into
unequal daughter droplets of calibrated size, and sorting. A
sampler which uses a combination of channel geometry and
laser forcing is shown in Fig. 5. We see in it drops that are
longer than the channel width and that arrive at a symmetric
bifurcation, carried by the oil phase. At the bifurcation, the
drops divide into two parts whose lengths in the daughter
channels we label L1 and L2. We are interested in the ratio l ;
S(L12L2)/(L1 + L2)T which yields l = 0 for symmetric drops
and l = 1 for complete sorting. The brackets S T here denote
an average over several drops.
In the absence of the laser (Fig. 5a), we measure l = 0.022 ¡
0.01 for our microchannel, corresponding to a slight asymmetry in the microfabrication. When the laser is applied in
front of one of the two exits, the water–oil interface is
asymmetrically blocked at the laser position for a time tb,
while the other side continues to flow (Fig. 5b). After tb, both
sides of the drop continue forward into their respective
channels, but the retardation of the right hand droplet tip
produces an asymmetry in the breaking, measured by an
increase in l. Since the blocking time tb increases with the laser
power, so does the asymmetry in the division, as shown in
Fig. 5c. (See supporting video 3 in the ESI{.)
Above a critical power (approximately 100 mW for the
present configuration), the drop does not divide but is always
diverted into the opposite branch. This sorting operation may
be understood by considering the length of the droplet
upstream of the laser. If the upstream length decreases below
a critical size of approximately the channel width (corrected by
the displacement of the laser with respect to the channel
center), the drop takes a circular shape and loses contact with
the right hand wall. In this case, a tunnel opens for the oil to
flow between the drop and the wall and the drop does not
divide anymore but is pushed into the left hand channel.
Unequal droplet splitting may be achieved through passive
techniques, for example by varying the downstream resistance
1032 | Lab Chip, 2007, 7, 1029–1033
Fig. 5 A droplet sampler: (a) without laser forcing, a drop at a
bifurcation divides into approximately equal daughter droplets. (b) By
controlling the laser power (marked with a white circle, here P =
93 mW), we control the pinning time of one side of the interface and
thus the asymmetry of the division. Main channel width is 200 mm and
the time between images is 0.2 s. Operating conditions are Qwater =
0.02 mL min21, Qoil = 0.2 mL min21, and v0 = 5.2 mm. (c) Daughter
droplet size dependence on laser power (l = 0 is for symmetric drops
and l = 1 is for the sorter). The dashed line corresponds to the mean
value of l in the absence of the laser.
to the flow in simple cases.10,20 However, the optical actuation
adds an active component to the control of each droplet. It
thus provides an additional control parameter that can be used
in conjunction with passive control, independently of the
downstream conditions or of the microsystem’s complexity.
5. Generality and optimization
Our approach to controlling microfluidic droplets relies on alloptical techniques which have been greatly developed in recent
years in connection with microfluidic devices.21–25 Indeed,
optical trapping has become a standard tool in biophysics26
and holographic22 and generalized phase contrast25 methods
now allow a single laser to be divided into many spots which
can be independently manipulated. The application of beam
division techniques to thermocapillary control should be
relatively straight-forward and it will allow the parallel
implementation of many independent building blocks in a
complex network of channels. For instance, many valves,
fusers, and sorters may be operated independently through the
implementation of holographic division of the laser or by
sweeping a single beam with a galvanometric mirror.
Furthermore, a judicious choice of a laser wavelength and
fluid combinations can improve the efficiency of the approach.
In this regard, current experiments have reproduced the above
results with an infrared laser which acts directly on the water
molecules, allowing us to work without the need for an
absorbing dye.
Moreover, the physical scaling laws for this forcing
technique are favourable to further miniaturization, since the
technique takes advantage of the dominance of surface
effects in microfluidics. The force produced by the thermocapillary flow was theoretically found to scale as 1/R, where
R represents the in-plane radius of curvature of the drop
at the hot spot.12 The force is therefore expected to increase
as the drop size decreases, as long as the local heating
hypothesis may be maintained. This scaling may be used to
optimize the performance of the system, for example by
using channels with a variable width or by placing the laser at
the position with highest drop curvature. Such optimization
should allow the implementation of the devices with minimal
laser power, further promoting parallelization and portability.
The response time should also scale favourably with
miniaturization since it is limited by the heat and viscous
diffusion processes and thermal inertia. The latter decreases
as the cube of the length scale, and is therefore negligibly
small in microchannels. Moreover, the viscous diffusion time
(tvisc y ,2/n, where , . 30 mm is a typical length scale and n .
1026 m2 s21 is the kinematic viscosity of water) and the
thermal diffusion time (tth y ,2/D, where D . 1027 m2 s21 is a
typical thermal diffusion coefficient) are both on the order of a
few ms, indicating that droplet manipulation at the kHz range
may be possible.
Finally, since the method requires no moving parts or
special microfabrication, the forcing is reconfigurable in realtime and may be adapted to many different microchannel
geometries. Owing to its flexibility and scalability, our optical
approach offers a complete toolbox for droplet based lab-ona-chip applications.
Acknowledgements
Useful discussions are acknowledged with François Gallaire
and Régis Wunenburger. This work was partially funded by
the
CNRS
Projet
interdisciplinaire
de
recherche
‘‘Microfluidique et Microsystèmes Fluidiques’’, the Conseil
Régional d’Aquitaine (Université Bordeaux 1 group), and the
convention X-DGA (Ecole Polytechnique group).
References
1 T. M. Squires and S. R. Quake, Microfluidics: Fluid physics at the
nanoliter scale, Rev. Mod. Phys., 2005, 77(3), 977–1026.
2 J. Atencia and D. J. Beebe, Controlled microfluidic interfaces,
Nature, 2005, 437, 648–655.
3 A. Günther and K. F. Jensen, Multiphase microfluidics: from flow
characteristics to chemical and material synthesis, Lab Chip, 2006,
6, 1487–1503.
4 H. Song, D. L. Chen and R. F. Ismagilov, Reactions in droplets in
microfluidic channels, Angew. Chem., Int. Ed., 2006, 45,
7336–7356.
5 T. Thorsen, R. W. Roberts, F. H. Arnold and S. R. Quake,
Dynamic pattern formation in a vesicle-generating microfluidic
device, Phys. Rev. Lett., 2001, 86(18), 4163–4166.
6 S. L. Anna, N. Bontoux and H. A. Stone, Formation of dispersions
using ‘flow-focusing’ in microchannels, Appl. Phys. Lett., 2003,
82(3), 364–366.
7 R. Dreyfus, P. Tabeling and H. Willaime, Ordered and disordered
patterns in two phase flows in microchannels, Phys. Rev. Lett.,
2003, 90, 144505.
8 M. Chabert, K. D. Dorfman, P. de Cremoux, J. Roeraade and
J.-L. Viovy, Automated microdroplet platform for sample
manipulation and polymerase chain reaction, Anal. Chem., 2006,
78, 7722–7728.
9 K. Ahn, J. Agresti, H. Chong, M. Marquez and D. Weitz,
Electrocoalescence of drops synchronized by size-dependent flow
in microfluidic channels, Appl. Phys. Lett., 2006, 88, 264105.
10 D. R. Link, S. L. Anna, D. A. Weitz and H. A. Stone,
Geometrically mediated breakup of drops in microfluidic devices,
Phys. Rev. Lett., 2004, 92(5), 054503.
11 K. Ahn, C. Kerbage, T. Hynt, R. M. Westervelt, D. R. Link and
D. A. Weitz, Dielectrophoretic manipulation of drops for highspeed microfluidic sorting devices, Appl. Phys. Lett., 2006, 88,
024104.
12 C. N. Baroud, J.-P. Delville, F. Gallaire and R. Wunenburger,
Thermocapillary valve for droplet production and sorting,
Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 75(4),
046302.
13 C. Priest, S. Herminghaus and R. Seemann, Controlled electrocoalescence in microfluidics: Targeting a single lamella, Appl. Phys.
Lett., 2006, 89, 134101.
14 A. A. Darhuber and S. M. Troian, Principles of microfluidic
actuation by modulation of surface stresses, Annu. Rev. Fluid
Mech., 2005, 37, 425–455.
15 N. O. Young, J. S. Goldstein and M. J. Block, The motion of
bubbles in a vertical temperature gradient, J. Fluid Mech., 1959, 6,
350–356.
16 B. Berge, O. Konovalov, J. Lajzerowicz, A. Renault, J. P. Rieu,
M. Vallade, J. Als-Nielsen, G. Grübel and J. F. Legrand,
Melting of short 1-alcohol monolayers on water: Thermodynamics and X-ray scattering studies, Phys. Rev. Lett., 1994, 73(12),
1652–1655.
17 E. Sloutskin, C. D. Bain, B. M. Ocko and M. Deutsch, Surface
freezing of chain molecules at the liquid–liquid and liquid–air
interfaces, Faraday Discuss., 2005, 129, 1–14.
18 J. Bibette, D. C. Morse, T. A. Witten and D. A. Weitz, Stability
criteria for emulsions, Phys. Rev. Lett., 1992, 69, 2439–2442.
19 Y. Amarouchene, G. Cristobal and H. Kellay, Noncoalescing
drops, Phys. Rev. Lett., 2001, 87, 206104.
20 L. Ménétrier-Deremble and P. Tabeling, Droplet breakup in
microfluidic junctions of arbitrary angles, Phys. Rev. E: Stat.,
Nonlinear, Soft Matter Phys., 2006, 74, 035303.
21 A. Terray, J. Oakey and D. W. M. Marr, Microfluidic control
using colloidal devices, Science, 2002, 296, 1841–1844.
22 D. G. Grier, A revolution in optical maniupulation, Nature, 2003,
424, 810–816.
23 M. P. MacDonald, G. C. Spalding and K. Dholakia, Microfluidic
sorting in an optical lattice, Nature, 2003, 426, 421–424.
24 D. R. Burnham and D. McGloin, Holographic optical trapping of
aerosol droplets, Opt. Express, 2006, 14(9), 4176–4182.
25 I. R. Perch-Nielsen, P. J. Rodrigo, C. A. Alonzo and J. Glückstad,
Autonomous and 3d real-time multi-beam manipulation in a
microfluidic environment, Opt. Express, 2006, 14(25),
12199–12205.
26 J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham and
J. Käs, The optical stretcher - a novel, noninvasive tool to
manipulate biological materials, Biophys. J., 2001, 81, 767–784.
Lab Chip, 2007, 7, 1029–1033 | 1033
98
Chapter 4
Recent and future work
(c)
99
Below, some “other”, newer, projects will be described. We will begin with a description of a study on tip-streaming that was undertaken with Simon Molesin, an undergraduate student at Ecole Polytechnique during his “stage d’option”. We will end with
the current work that is underway at our lab and whose results are not yet in publishable
form.
4.1
Tip-Streaming
This work is still ongoing. It was started with the “stage” of Simon Molesin
at LadHyX (in 2006-2007) and is being continued with Thomas Dubos.
The production of singular tips on drops has received significant attention, which
was increased recently by some detailed work on interfaces submitted to strong shear,
whether through “selective withdrawal” [35, 39] or through air entrainment [84]. These
singular structures are produced on an otherwise quiescent layer of liquid, which makes
their observation and measurements possible. On the other hand, the production of
tips on moving droplets has a long history, since it was first observed by G.I. Taylor
who described the stretching of a drop submitted to shear and streaming smaller drops
as “tip streaming” in 1934 [117]. More recently, this phenomenon was studied from a
phenomenological point of view by de Bruijn, who noted that tip streaming only appeared
for intermediate surfactant concentrations, which suggests that surfactant must play an
important role in the process [43]. Later studies on tip-streaming were mainly numerical
in nature [50, 16], although the phenomenon has been reported experimentally in recent
papers [7, 8, 53].
However, detailed studies remained difficult. The challenge arises, in the case of
experiments, in the difficulty of making detailed quantitative measurements on the drops.
Indeed, most experiments to date consist of qualitative observations of a single drop in
an externally imposed flow, such as a four roll mill or a Couette flow [43, 89, 105]. In
the case of numerical simulations, studies of tip streaming are complicated by the need
to follow of a free surface coupled with several nonlinear equations which account for the
surfactant transport, adsorption/desorption dynamics, and the relation between surface
concentration and surface tension [94, 50, 16].
The experimental setup we used consisted of a circular shaped PDMS microchannel.
The channel had four PDMS walls, with a cross-section width w = 500 µm and height
h = 35 µm. The outer radius of the circular channel was 2 mm, allowing observations to
be done using an inverted microscope (×4 to ×15 magnification) and a fast camera at
1024 × 1024 pixel resolution, as shown in Fig. 4.1a.
Water and paraffin oil are forced through syringe pumps into the microchannel,
where drops of water in oil are produced in a continuous fashion at a T-junction. Several
hundred drops were studied in this fashion while varying the bulk surfactant concentration
(c), the total flow rate (Q), and the drop size. The oil viscosity (µo ) was measured to
be 0.21 Pa s, yielding a Reynolds number below below 10−2 for all of our experimental
conditions. Surfactant (Span80) was added to the oil at three concentrations: 0.05, 0.15,
and 0.5% wt/wt. Finally, the surface tension for the three concentrations was measured
using the pendant drop technique, giving γ = 16 · 10−3 , 9 · 10−3 , and 5 · 10−3 N/m
100
(a)
!
Flow
r
(b)
(c)
Center of mass : dt=0.017s
Drop : dt=0.35s
Figure 4.1: (a) Experimental snapshot of the circular channel with several drops flowing
clockwise. The channel width is 500 µm. (b) Reconstituted history of a single drop
showing the shape and centroid position at regular time intervals. (c) Detail photograph
showing two tips streaming from the rear of a drop. Experimental conditions: Q =
5.2 µL/min and c = 0.05%.
respectively for the three concentrations. The surface tension in the absence of surfactant
was measured to be 25 · 10−3 N/m.
Owing to the large width/height aspect ratio, one may use the potential flow theory
to write the velocity profile of the base flow as U = φ/r, where φ is a constant that depends
on the total flow rate and r is the radial position measured from the center of the circle.
Note that this approximation applies to the velocity averaged across the depth of the
channel and is valid a distance of order h from the inner and outer walls. The velocity
profile in the direction normal to the plane of the channel is expected to be parabolic.
Furthermore, since the channel width (w = 500 µm) is small compared with the mean
radius (R̄ = 1750 µm), one may approximate the shear rate as a constant to first order.
PIV measurements were used to validate the flowrate estimates.
The effect of the shear flow can be observed in the snapshot of Fig. 4.1a. The drops
start out nearly circular (first drop upstream) and their shape begins to deviate as they
enter the bend (second and third drops). During this period, sharp corners appear on
the rear of the drop, which may also display a straight interface (e.g. fourth drop). We
101
associate this “rigidity” with the continuous deposition of surfactant molecules on the rear
interface. Indeed, the symmetry between the leading and trailing edges is broken by the
no-slip condition at the upper and lower walls, so the stationary oil layer continuously
sweeps surfactant molecules to the rear of the drop. The drops continuously elongate
until a critical point at which they change their orientation and position, aligning their
major axis tangent to the channel and migrating towards the inner channel wall (most
downstream drop).
A reconstructed history of a single drop is shown at regular time intervals in
Fig. 4.1b. The position of the centroid (red circles) shows that the drop accelerates
and its center migrates towards the inner wall. These shape and velocity transitions
are accompanied by the production of one or more tails at the rear of the drop, which
then break up into small droplets behind the mother drop (Fig. 4.1c). The droplets thus
produced are polydisperse but a size histogram displays a large peak at the limit of our
optical resolution, implying the presence of a large population below a diameter of about
1 µm. This tip streaming only appears for large enough total flow rate and in the presence of surfactant. Experiments carried out without surfactant showed no tip streaming,
while those with a high surfactant concentration showed reduced tails and smaller drop
acceleration.
5
10
1.2
(a)
1.15
8
Area (µm2)
Speed (mm/s)
x 10
(b)
9
7
6
5
1
0.95
4
3
!!/2
1.1
1.05
0
!/2
0.9
!!/2
!
Azimuthal position (rad)
0
!/2
Azimuthal position (rad)
!
Figure 4.2: Typical evolution for one drop: (a) the velocity as a function of azimuthal
position. The dashed line gives the fluid’s mean nominal velocity. (b) Drop area as
measured (o) and as computed (x) from the velocity using Bretherton’s law. Experimental
condition : c = 0.05 %, Q = 5.2 µL/min.
Several quantities go through large changes during the transition to tip-streaming.
In particular, the velocity history of a drop is shown in Fig. 4.2a as a function of the
azimuthal position of the centroid (θ). The drop initially travels slower than the mean
velocity of the oil (marked by the dashed line) but it accelerates to reach a value about
twice the mean fluid velocity by the end of the channel.
The increase in velocity is also associated with an increase in the apparent surface
area of the drop (Fig. 4.2b), which can be understood by considering the thickness of
the lubricating film on the upper and lower channel walls: the thickness (e) of this film
may be estimated using Bretherton’s law [24] which states that the film left behind an
advancing air finger increases as e/h ∼ Ca2/3 . This scaling was verified by using the
initial thickness (e0 ) of the film as a free fitting parameter and computing the predicted
surface area of the drop as A = A0 [h − 2e0 ]/[h − 2e0 (V /V0 )2/3 ]. A comparison of the
measured area with the prediction for A shows good agreement, although a time lag is
often observed. The initial film thickness was thus found to be approximately 5 − 8 µm
for the different experimental condition.
102
By finding the azimuthal position of maximum radial extent, we can define a position
for the destabilization. This is shown in Fig. 4.3(a) where we have plotted the normalized
relative radial extension (∆R = (dr−dr0 )/dr) of different drops as a function of azimuthal
position. We observe that ∆R follows the same evolution for all experimental conditions.
The position for destabilization can then be measured for the different control parameters,
as shown in Fig. 4.3. We observe that the critical angle θc depends strongly on the drop
size and on the total flowrate, but does not show a strong dependence on the surfactant
concentration.
!R=(dr!dr0)/dr
0.4
(a)
0.3
0.2
0.1
0
!0.1
!1.57
0
1.57
3.14
Azimuhtal position (rad)
3
3
2
!c
c
!
(c)
(b)
2.5
2
1.5
1
90
1
0
100
110
120
130
140
150
Equivalent radius (!m)
!1
4
5
6
7
8
Flow rate (µL/min)
Figure 4.3: (a) Relative radial stretching as a function of azimuthal position. The same
evolution is observed for all the drops, regardless of experimental conditions. (b) Critical
azimuthal position for destabilization (θc ) as a function of drop size for two flow rates (◦)
Q = 6.2 µL/min and (∗) Q = 8.2 µL/min). (c) Variations in θc with flow rate for a given
drop surface area (A = 82.103 µm2 ± 7%).
Two capillary numbers can be written which describe the balance between viscous
and capillary effects on a drop. The first one accounts for the fact that the drop is
subjected to a parabolic profile in the thickness of the channel. This number, which we
write Ca⊥ , may be written as:
1/3
a µo Vo
a µo Vo
,
=
Ca =
γ e
h
γ
⊥
(4.1)
where µo is the oil viscosity, γ is a characteristic interfacial tension value (e.g. 20 mN/m),
Vo is the drop velocity at the entrance, and a is the drop size, obtained from its apparent
area.
The value of this capillary number at the moment of destabilization is plotted
on Fig. 4.4a (?) as a function of the drop size, for a given of flow rate and surfactant
concentration. This value is seen to monotonically increase with drop size.
This capillary number must be modified, however, by the shear G experienced by
the drop in the plane of the channel, which is imposed externally by the circular geometry.
This second capillary number can be written as
103
µo Ga
∆R,
(4.2)
γ
where G is the shear rate from the base flow, ∆R is the relative radial extension of the
drop, and α a free parameter which will be used to compare the two capillary numbers
(α ' 90 in our case).
Cak = α
0.4
0.4
0.3
Total Ca
Cac
Capillary number
0.3
0.2
0.2
Ca!
0.1
0
90
0.1
Ca//
100
110
120
130
140
Equivalent radius (µm)
150
160
170
0
4
5
6
7
Flow rate (µL/min)
8
Figure 4.4: (a) Contribution of the different components of capillary number at destabilization. Experimental conditions : Flow rate 8 µ L/min - Surfactant concentration
= 0.05% (b) Critical value of the capillary number at the destabilization. The error
bars represent the amplitude of values fixing flow rate and surfactant concentration. The
different values of the surfactant concentration are 0.05%(o), 0.15%(x) and 0.5%(+).
In this way, one may write a total capillary number as the sum Ca = Cak + Ca⊥ .
Drops were found to destabilize for a constant value of Ca independent of their drop size
(for a . 150 µm), as shown in Fig. 4.4(left). We see that Ca takes on a constant critical
value, independently of flow rate, surfactant concentration, or drop size (Fig 4.4(right)).
The significance of this “total capillary number” is still not fully understood, but
these experiments represent a clear example where microfluidics offers a unique tool to
access fluid dynamical situations which would otherwise be very difficult to study.
4.2
Flows in networks
This work is currently underway, in collaboration with Yu Song (phd student)
and Michaël Baudoin (post-doc), while closely working with Paul Manneville.
The results of Chap. 2 lead to the natural desire to look at several generations of
bifurcations rather than limiting oneself to flow through a single bifurcation. However,
flows in complex networks, as the one shown in Fig. 4.5(left), require mathematical tools
distinct from the ones developed in our previous work. The dynamical description of the
previous sections must now be replaced with the statistical description of a flow through
a branched structure.
The flow of plugs of liquid pushed by a mean air flux is currently being studied
at LadHyX, as shown in Fig. 4.5(right). The questions we wish to pursue are related
to the stability of the flow distribution through the network, i.e. whether the nonlinearities introduced by the drops will lead to strongly inhomogeneous flow through the
104
Figure 4.5: (left) An early realization of a network of microchannels at LadHyX. (right)
Microscopy image of a network containing a random distribution of plugs.
whole branched structure. Preliminary results confirm that the downstream boundary
conditions play a major role in determining the flow paths and may have to be taken into
account in the theoretical modeling of the network.
4.3
Manipulation of single cells in drops
It is only fitting to end this document with the application of droplet methods to biological
experiments, since we started by stating the major impact that microfluidics could have
on biology and chemistry. This for us will consist of very recent but mainly future work,
where we explore the manipulation of single cells (or a small population of cells) inside a
droplet. Contrary to the experiments discussed in the previous parts of the manuscript,
the aim here is neither to explore toy models nor to develop new tools. The purpose of
these studies is to shed new light on biological mechanisms by using the tools that we have
already developed. These biological projects are very exciting but involve a significant
effort to develop the biological know-how in the ladhyx group.
For this, several collaborations have been started with biologists and biophysicists,
for example at the “Laboratoire d’Optique et Biosciences” (LOB) at Ecole Polytechnique,
where a project was started with Antigoni Alexandrou to run experiments on a large
number of cells, by flowing the cells in drops in series. An early image of a cell in a drop
is shown in Fig. 4.6, where a red blood cell is followed in time as the drop that carries it
travels from left to right.
Figure 4.6: Superposition of eight images showing the evolution of a red blood cell in a
drop.
The current approach to biological experiments generally consists either in performing detailed and time consuming experiments on a single cell, or in working with tissues
105
where the number of cells is large. The gap that we hope to bridge with our project
is to be able to work with a large number of single cells, thus combining the statistical
power of tissue experiments with the detailed measurements of single cell experiments.
Microfluidics can allow such measurements since one may perform many copies of an
experiment in drops that follow each other, while varying the parameters slightly. By
performing experiments in this way, one can expect to provide a new type of information
on biological processes that can help in understanding the complex mechanisms at play
in biology.
106
Bibliography
[1] http://www.parl.clemson.edu/stare.
[2] Abkarian, M., Faivre, M., and Stone, H. High-speed microfluidic differential
manometer for cellular-scale hydrodynamics. Proc. Nat. Acad. Sci. 103, 3 (2006),
538–542.
[3] Adzima, B., and Velankar, S. Pressure drops for droplet flows in microfluidic
channels. J. of micromechanics and microengineering 16, 8 (2006), 1504 – 1510.
[4] Ahn, K., Agresti, J., Chong, H., Marquez, M., and Weitz, D. Electrocoalescence of drops synchronized by size-dependent flow in microfluidic channels.
Appl. Phys. Lett. 88 (2006), 264105.
[5] Ahn, K., Kerbage, C., Hynt, T., Westervelt, R., Link, D., and Weitz,
D. Dielectrophoretic manipulation of drops for high-speed microfluidic sorting devices. Appl. Phys. Lett. 88 (2006), 024104.
[6] Ajaev, V. S., and Homsy, G. Modeling shapes and dynamics of confined bubbles. Ann. Rev. Fluid Mech. 38, 1 (2006), 277–307.
[7] Anna, S., Bontoux, N., and Stone, H. Formation of dispersions using ’flowfocusing’ in microchannels. Appl. Phys. Lett. 82, 3 (2003), 364–366.
[8] Anna, S., and Mayer, H. Microscale tipstreaming in a microfluidic flow focusing
device. Physics of Fluids 18, 12 (2006), 121512.
[9] Aubert, C., and Colin, S. High-order boundary conditions for gaseous flows in
rectangular microducts. Nanoscale and Microscale Thermophysical Engineering 5,
1 (2001), 41–54.
[10] Barbier, V., Willaime, H., Tabeling, P., and Jousse, F. Producing
droplets in parallel microfluidic systems. Phys. Review E 74, 4 (2006), 046306.
[11] Baroud, C., de Saint Vincent, M., and Delville, J.-P. An optical toolbox
for total control of droplet microfluidics. Lab Chip 7 (July 2007), 1029–1033.
[12] Baroud, C., Delville, J., Gallaire, F., and Wunenburger, R. Thermocapillary valve for droplet production and sorting. Phys. Rev. E 75, 4 (2007),
046302.
107
[13] Baroud, C., Okkels, F., Ménétrier, L., and Tabeling, P. Reactiondiffusion dynamics: confrontation between theory and experiment in a microfluidic
reactor. Phys. Rev. E 67 (July 2003), 060104(R).
[14] Baroud, C., Tsikata, S., and Heil, M. The propagation of low-viscosity
fingers into fluid-filled, branching networks. J. Fluid Mech. 546 (2006), 285–294.
[15] Baroud, C., Wang, X., and Masson, J.-B. Collective behavior during the
exit of a wetting liquid through a network of channels. J. Colloid and Interface Sci.
326 (October 2008), 445–450.
[16] Bazhlekov, I., Anderson, P., and Meijer, H. Numerical investigation of
the effect of insoluble surfactants on drop deformation and breakup in simple shear
flow. J. Colloid Interface Sci. 298 (2006), 369–394.
[17] Berthier, E., and Beebe, D. Flow rate analysis of a surface tension driven
passive micropump. Lab on a Chip 7, 11 (2007), 1475–1478.
[18] Bibette, J., Morse, D., Witten, T., and Weitz, D. Stability criteria for
emulsions. Phys. Rev. Lett. 69 (1992), 2439–2442.
[19] Bico, J., and Quéré, D. Falling slugs. J. Coll. and Interface Science 243 (2001),
262–264.
[20] Bohn, S., Andreotti, B., Douady, S., Munzinger, J., and Couder, Y.
Constitutive property of the local organization of leaf venation networks. Phys.
Rev. E 65, 6 (Jun 2002), 061914.
[21] Boos, W., and Thess, A. Thermocapillary flow in a hele-shaw cell. J. Fluid
Mech. 352 (1997), 305–320.
[22] Braun, D., and Libchaber, A. Trapping of dna by thermophoretic depletion
and convection. Phys. Rev. Lett. 89 (2002), 188103.
[23] Bremond, N., Thiam, A., and Bibette, J. Decompressing emulsion droplets
favors coalescence. Physical Review Letters 100, 2 (2008), 024501.
[24] Bretherton, F. The motion of long bubbles in tubes. J. Fluid Mech. 10 (1961),
166–188.
[25] Brouzes, E., Branciforte, J., Twardowski, M., Marran, D., Suo, Y.,
Charles, Y., Boitard, L., Perrimon, N., and Link, D. Droplet-based highthroughput live/dead cell assay. In Proceedings of Micro-TAS 2006 (2006).
[26] Bryant, S., Mellor, D., and Cade, C. Physically representative network
models of transport in porous media. AIChE Journal 39, 3 (1993), 387–396.
[27] Burnham, D., and McGloin, D. Holographic optical trapping of aerosol
droplets. Optics Express 14, 9 (2006), 4176–4182.
108
[28] Burnham, D., Wright, G., Read, N., and McGloin, D. Holographic and
single beam optical manipulation of hyphal growth in filamentous fungi. Journal
of Optics A: Pure and Applied Optics 9, 8 (2007), S172–S179.
[29] Bush, J. The anomalous wake accompanying bubbles rising in a thin gap: a
mechanically forced marangoni flow. J. Fluid Mech. 352 (1997), 283–303.
[30] Calderon, A., and Fowlkes, B. Bull, J. Bubble splitting in bifurcating
tubes: a model study of cardiovascular gas emboli transport. J. Appl. Physiol. 99
(2005), 479–487.
[31] Cassidy, K., Gavriely, N., and Grotberg, J. Liquid plug flow in straight
and bifurcating tubes. J. Biomech. Eng. 123 (2001), 580–589.
[32] Chabert, M., Dorfman, K., de Cremoux, P., Roeraade, J., and Viovy,
J.-L. Automated microdroplet platform for sample manipulation and polymerase
chain reaction. Anal. Chem. 78 (2006), 7722–7728.
[33] Chabert, M., Dorfman, K. D., and Viovy, J. Droplet fusion by alternating
current (AC) field electrocoalescence in microchannels. Electrophoresis 26 (2005),
3706–3715.
[34] Cicciarelli, B., Elia, J., Hatton, T., and Smith, K. Temperature dependence of aggregation and dynamic surface tension in a photoresponsive surfactant
system. Langmuir 23, 16 (2007), 8323–8330.
[35] Cohen, I., and Nagel, S. Scaling at the selective withdrawal transition through
a tube suspended above the fluid surface. Phys. Rev. Lett. 88, 7 (2002), 074501.
[36] Constantin, P., Dupont, T., Goldstein, R., Kadanoff, L., and Shelley,
M. Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E 47, 6 (1993),
4169–4181.
[37] Cordero, M., Burnham, D., Baroud, C., and McGloin, D. Thermocapillary manipulation of droplets using holographic beam shaping: Microfluidic pin
ball. Appl. Phys. Lett. 93, 3 (2008), 034107.
[38] Cordero, M., Verneuil, E., and Baroud, C. Optical forcing of microdrops
: Flows and temperature field characterization. In Proceedings of MicroTAS 2007
(2007).
[39] Courrech du Pont, S., and Eggers, J. Sink flow deforms the interface
between a viscous liquid and air into a tip singularity. Phys. Rev. Lett. 96 (2006),
034501–1–4.
[40] Cristobal, G., Benoit, J., Joanicot, M., and Ajdari, A. Microfluidic
bypass for efficient passive regulation of droplet traffic at a junction. Appl. Phys.
Lett. 89 (2006), 034104.
[41] Darhuber, A., and Troian, S. Principles of microfluidic actuation by modulation of surface stresses. Annu. Rev. Fluid Mech. 37 (2005), 425–455.
109
[42] Darhuber, A., Valentino, J., Davis, J., Troian, S., and Wagner, S.
Microfluidic actuation by modulation of surface stresses. Appl. Phys. Letters 82, 4
(January 2003), 657–659.
[43] de Bruijn, R. Tipstreaming of drops in simple shear flows. Chem. Eng. Sci. 48,
2 (1993), 277–294.
[44] de Gennes, P. Wetting: statics and dynamics. Rev. Mod. Phys. 57, 3 (1985),
827–863.
[45] Dollet, B., van Hoeve, W., Raven, J., Marmottant, P., and Versluis,
M. Role of the channel geometry on the bubble pinch-off in flow-focusing devices.
Physical Review Letters 100, 3 (2008), 034504.
[46] Dreyfus, R., Tabeling, P., and Willaime, H. Ordered and discordered
patterns in two phase flows in microchannels. Phys. Rev. Lett. 90 (April 2003),
144505.
[47] Duclaux, V., Clanet, C., and Quéré, D. The effects of gravity on the
capillary instability in tubes. J. Fluid Mech. 556 (2006), 217–226.
[48] Duhr, S., Arduini, S., and Braun, D. Thermophoresis of dna determined by microfluidic fluorescence. Eur. Phys. J. E 15 (2004), 277–286. DOI
10.1140/epje/i2004-10073-5.
[49] Eastoe, J., and Dalton, J. Dynamic surface tension and adsorption mechanisms of surfactants at the air-water interface. Adv. Colloid Interface Sci. 85 (2000),
103–144.
[50] Eggleton, C., Tsai, T.-M., and Stebe, K. Tip streaming from a drop in the
presence of surfactants. Phys. Rev. Lett. 87 (2001), 048302–1–4.
[51] Engl, W., Roche, M., Colin, A., Panizza, P., and Ajdari, A. Droplet
Traffic at a Simple Junction at Low Capillary Numbers. Physical Review Letters
95, 20 (2005), 208304.
[52] Espinosa, F., and Kamm, R. Bolus dispersal through the lungs in surfactant
replacement therapy. J. Appl. Physiol. 86, 1 (1999), 391–410.
[53] Fernandez, J., and Homsy, G. Chemical reaction-driven tip-streaming phenomena in a pendant drop. Phys. Fluids 16, 7 (2004), 2548–2555.
[54] Ferri, J., and Stebe, K. Which surfactants reduce surface tension faster? a
scaling argument for diffusion-controlled adsorption. Adv. Colloid and Interface
Science 85 (2000), 61–97.
[55] Fuerstman, M., Garstecki, P., and Whitesides, G. coding/decoding and
reversibility of droplet trains in microfluidic networks. Science 315 (2007), 828–832.
110
[56] Fuerstman, M., Lai, A., Thurlow, M., Shevkoplyas, S., Stone, H., and
Whitesides, G. The pressure drop along rectangular microchannels containing
bubbles. Lab on a Chip 7, 11 (2007), 1479–1489.
[57] Gallaire, F., and Baroud, C. Marangoni induced force on drop in a microchannel. In Preparation (2007).
[58] Garstecki, P., Stone, H., and Whiteseides, G. Mechanism for flow-rate
controlled breakup in confined geometries: A route to monodisperse emulsions.
Phys. Rev. Lett. 94 (2005), 164501.
[59] Grier, D. A revolution in optical maniupulation. Nature 424 (Aug. 2003), 810–
816.
[60] Guillot, P., Colin, A., Utada, A., and Ajdari, A. Stability of a jet in
confined pressure-driven biphasic flows at low reynolds numbers. Phys. Rev. Lett
99 (2007), 104502.
[61] Günther, A., and Jensen, K. Multiphase microfluidics: from flow characteristics to chemical and material synthesis. Lab Chip 6 (2006), 1487–1503.
[62] Hazel, A. L., and Heil, M. The steady propagation of a semi-infinite bubble
into a tube of elliptical or rectangular cross-section. J. Fluid Mech. 470 (2002),
91–114.
[63] Hoffman, R. A study of the advancing interface. J. Colloid and Interface Science
50, 2 (75), 228–241.
[64] Hu, Y., and Lips, A. Estimating surfactant surface coverage and decomposing
its effect on drop deformation. Phys. Rev. Lett. 91 (2003), 044501.
[65] Huang, L., Cox, E., Austin, R., and Sturm, J. Continuous particle separation through deterministic lateral displacement. Science 304 (2004), 987–990.
[66] Hudson, S. D., Cabral, J. T., William J. Goodrum, J., Beers, K. L.,
and Amis, E. J. Microfluidic interfacial tensiometry. Appl. Phys. Lett. 87, 8
(2005), 081905.
[67] Ismagilov, R., Stroock, A., Kenis, P., Whitesides, G., and Stone, H.
Experimental and theoretical scaling laws for transverse diffusive broadening in
two-phase laminar flows in microchannels. Appl. Phys. Lett. 76, 17 (April 2000),
2376–2378.
[68] Jeffries, G., Kuo, J., and Chiu, D. Controlled Shrinkage and Re-expansion of
a Single Aqueous Droplet inside an Optical Vortex Trap. Journal of physical chemistry. B, Condensed matter, materials, surfaces, interfaces, & biophysical chemistry
111, 11 (2007), 2806–2812.
[69] Jin, F., Gupta, N., and Stebe, K. The detachment of a viscous drop in
a viscous solution in the presence of a soluble surfactant. Physics of Fluids 18
(2006), 022103.
111
[70] Joanicot, M., and Ajdari, A. Droplet control for microfluidics. Science 309
(2005), 887–888.
[71] Jung, B., Bharadwaj, R., and Santiago, J. Thousandfold signal increase
using field-amplified sample stacking for on-chip electrophoresis. Electrophoresis 24
(2003), 3476–3483.
[72] Khan, S., Günther, A., Schmidt, M., and Jensen, K. Microfluidic synthesis
of colloidal silica. Langmuir 20, 20 (2004), 8604–8611.
[73] Kopp, M., de Mello, A., and Manz, A. Chemical amplification: Continuousflow pcr on a chip. Science 280 (May 1998), 1046–1048.
[74] Kotz, K., Noble, K., and Faris, G. Optical microfluidics. Appl. Phys. Lett.
85, 13 (2004), 2658–2660.
[75] Lajeunesse, E., and Homsy, G. Thermocapillary migration of long bubbles in
polygonal tubes. ii. experiments. Phys. Fluids 15 (2003), 308–314.
[76] Laser, D., and Santiago, J. A review of micropumps. J. Micromech. Microeng.
14 (2004), R35–R64.
[77] Laval, P., Salmon, J., and Joanicot, M. A microfluidic device for investigating crystal nucleation kinetics. Journal of Crystal Growth 303, 2 (2007), 622–628.
[78] Lee, J. N., Park, C., and Whitesides, G. Solven compatibility of
poly(dimethylsiloxane)-based microfluidic devices. Anal. Chem. 75 (2003), 6544–
6554.
[79] Lenormand, R. Pattern growth and fluid displacements through porous media.
Physica 140A (1986), 114–123.
[80] Lenormand, R., and Zarcone, C. Role of roughness and edges during imbibition in square capillaries. SPE (1984).
[81] Levich, V. Physicochemical Hydrodynamics. Prentice Hall, Englewood Clifs, N.J.,
1962.
[82] Link, D., Anna, S., Weitz, D., and Stone, H. Geometrically mediated
breakup of drops in microfluidic devices. Phys. Rev. Lett. 92, 5 (2004), 054503.
[83] Liu, R., Stremler, M., Sharp, K., Olsen, M., Santiago, J., Adrian,
R., Aref, H., and Beebe, D. Passive mixing in a three-dimensional serpentine
microchannel. Microelectromechanical Systems, Journal of 9, 2 (2000), 190–197.
[84] Lorenceau, E., Restagno, F., and Quéré, D. Fracture of a viscous liquid.
Phys. Rev. Lett. 90, 18 (May 2003), 184501.
[85] Madou, M. Fundamentals of microfabrication: The science of miniaturization,
second ed. CRC Press, 2002.
112
[86] Marmottant, P., and Hilgenfeldt, S. A bubble-driven microfluidic transport
element for bioengineering. Proc. Nat. Acad. Sci. 101, 26 (2004), 9523–9527.
[87] Maurer, J., Tabeling, P., Joseph, P., and Willaime, H. Second-order
slip laws in microchannels for helium and nitrogen. Phys. Fluids 15, 9 (2003),
2613–2621.
[88] Mazouchi, A., and Homsy, G. Thermocapillary migration of long bubbles in
cylindrical capillary tubes. Phys. Fluids 12, 3 (2000), 542–549.
[89] Mietus, W., Matar, O., Lawrence, C. J., and Briscoe, B. J. Droplet
deformation in con. Chem. Eng. Sci. 57 (2002), 1217–1230.
[90] Nadim, A., Borhan, A., and Haj-Hariri, H. Tangential stress and marangoni
effects at a fluid-fluid interface in a hele-shaw cell. J. Colloid and Interface Science
181 (1996), 159–164.
[91] Ody, C., Baroud, C., and de Langre, E. Transport of wetting liquid plugs
in bifurcating microfluidic channels. J. Colloid Interface Sci. 308 (Jan. 2007), 231–
238.
[92] Okkels, F., and Tabeling, P. Spatiotemporal Resonances in Mixing of Open
Viscous Fluids. Physical Review Letters 92, 3 (2004), 38301.
[93] Oron, A., Davis, S., and Bankoff, S. Long-scale evolution of thin liquid films.
Revs. Modern Physics 69, 3 (1997), 931 – 980.
[94] Park, C.-W., Maruvada, S., and Yoon, D.-Y. The influence of surfactant
on the bubble motion in hele-shaw cells. Phys. Fluids 6 (1994), 3267–3275.
[95] Pollack, M., Fair, R., and Shenderov, A. Electrowetting-based actuation
of liquid droplets for microfluidic applications. Appl. Phys. Lett. 77, 11 (2000),
1725–1726.
[96] Priest, C., Herminghaus, S., and Seemann, R. Controlled electrocoalescence
in microfluidics: Targeting a single lamella. Appl. Phys. Lett. 89 (Sept 2006),
134101.
[97] Quillet, C., and Berge, B. Electrowetting: A recent outbreak. Curr. Opin.
Colloid Interface Sci. 6, 1 (2001), 34–39.
[98] Renaudin, A., Tabourier, P., Zhang, V., Camart, J., and Druon, C.
SAW nanopump for handling droplets in view of biological applications. Sensors &
Actuators: B. Chemical 113, 1 (2006), 389–397.
[99] Ross, D., Gaitan, M., and Locascio, L. Temperature measurements in microfluidic systems using a temperature-dependent fluorescent dye. Anal. Chem. 73
(2001), 4117–4123.
113
[100] Saint-Jalmes, A., Zhang, Y., and Langevin, D. Quantitative description
of foam drainage: Transitions with surface mobility. Eur. Phys. J. E 15 (2004),
53–60.
[101] Sammarco, T., and Burns, M. Thermocapillary pumping of discrete drops in
microfabricated analysis devices. AIChE J. 45, 2 (1999), 350–366.
[102] Sarrazin, F., Loubière, K., Prat, L., Gourdon, C., Bonometti, T., and
Magnaudet, J. Experimental and numerical study of droplets hydrodynamics in
microchannels. AiChE J. 52 (2006), 4061–4070.
[103] Sarrazin, F., Prat, L., Di Miceli, N., Cristobal, G., Link, D., and
Weitz, D. Mixing characterization inside microdroplets engineered on a microcoalescer. Chem. Eng. Sci. 62 (2007), 1042–1048.
[104] Schindler, M., and Ajdari, A. Droplet traffic in microfluidic networks: A
simple model for understanding and designing. Physical Review Letters 100, 4
(2008), 044501.
[105] Sibillo, V., Pasquariello, G., Simeone, M., Cristini, V., and Guido,
S. Drop deformation in microconfined shear flow. Phys. Rev. Lett. 97, 5 (2006),
054502.
[106] Song, H., and Ismagilov, R. Millisecond kinetics on a microfluidic chip using
nanoliters of reagents. J. Am. Chem. Soc 125, 14 (2003), 613.
[107] Song, H., Tice, J., and Ismagilov, R. A microfluidic system for controlling
reaction networks in time. Angew. Chem., Int. Ed. 42, 7 (2003), 768–772.
[108] Squires, T., and Quake, S. Microfluidics: Fluid physics at the nanoliter scale.
Rev. Mod. Phys. 77, 3 (July 2005), 977–1026.
[109] Stark, J., and Manga, M. The motion of long bubbles in a network of tubes.
Transport in Porous Media 40 (2000), 201–218.
[110] Stone, H., Strook, A., and Ajdari, A. Engineering flows in small devices:
Microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36 (2004), 381–411.
[111] Strook, A., Dertinger, S., Ajdari, A., Mezic, I., Stone, H., and Whitesides, G. Chaotic mixer for microchannels. Science 295 (2002), 647–651.
[112] Sun, B., Lim, D., Kuo, J., Kuyper, C., and Chiu, D. Fast initiation of
chemical reactions with laser-induced breakdown of a nanoscale partition. Langmuir
20, 22 (2004), 9437–9440.
[113] Tan, Y.-C., Fisher, J., Lee, A., Cristini, V., and Lee, A. Design of
microfluidic channel geometries for the control of droplet volume, chemical concentration, and sorting. Lab Chip 4 (2004), 292–298.
[114] Tanner, L. The spreading of silicone oil drops on horizontal surfaces. J. Phys. D:
Appl. Phys. 12 (1979), 1473–1484.
114
[115] Tawfik, D., and Griffiths, A. Man-made cell-like compartments for molecular
evolution. Nature Biotech. 16 (1998), 652–656.
[116] Taylor, G. The viscosity of a fluid containing small drops of another fluid. Proceedings of the Royal Society of London. Series A 138 (1932), 41–48.
[117] Taylor, G. The formation of emulsions in definable fields of flow. Proceedings of
the Royal Society of London. Series A 146 (1934), 501–523.
[118] Taylor, G. Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10
(1961), 161–165.
[119] Velev, O., Prevo, B., and Bhatt, K. Onchip manipulation of free droplets.
Nature 426 (2003), 515–516.
[120] Vennemann, P., Kiger, K., Lindken, R., Groenendijk, B. C.,
Stekelenburg-de vos, S., Ten Hagen, T. L., Ursem, N. T., Poelmann,
R. E., Westerweel, J., and Hierck, B. P. In vivo micro particle image velocimetry measurements of blood-plasma in the embryonic avian heart. Journal of
Biomechanics 39, 7 (2006), 1191 – 1200.
[121] Verneuil, E., Cordero, M., Gallaire, F., and Baroud, C. Anomalous
thermocapillary force on microdroplet: Origin and magnitude. Submitted (2008).
[122] Weibel, E. R. The pathway for oxygen. Harvard U. press, Cambridge, MA., USA,
1984.
[123] White, F. Viscous fluid flow. McGraw Hill, 1991.
[124] Young, N., Goldstein, J., and Block, M. The motion of bubbles in a vertical
temperature gradient. J. Fluid Mech. 6 (1959), 350–356.
[125] Zheng, B., Tice, J., Roach, L., and Ismagilov, R. A droplet based, composite pdms/glass capillary microfluidic system for evaluating protein crystallization
conditions by microbatch and vapor-diffusion methods with on-chip x-ray diffraction. Angew. Chem. Int. Ed. 43, 19 (May 2004), 2508–2511.
[126] Zhu, X., and Kim, E. Microfluidic motion generation with acoustic waves. Sensors & Actuators: A. Physical 66, 1-3 (1998), 355–360.
115

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